#493796

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1804
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1804
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,5,10,7,13,26,12)(3,14,18,8,19,33,16,34)(4,20,21,41,24,55,22,54)(6,27,25,11,30,39,58,42)(9,35,36,57,38,29,37,23)(15,45,47,61,49,64,46,53)(17,32,52,63,28,40,50,44)(31,56,60,59,62,48,43,51);(1,3,7,19)(2,8)(4,15,24,49)(5,16,26,18)(6,28,30,17)(9,31)(10,14)(11,40)(12,33)(13,34)(20,53)(21,46,22,47)(23,56,57,48)(25,52,58,50)(27,44)(29,59,35,51)(32,42)(36,43)(37,60)(38,62)(39,63)(41,64)(45,54)(55,61);(1,4)(2,9)(3,15)(5,21)(6,29)(7,24)(8,31)(10,36)(11,41)(12,37)(13,38)(14,43)(16,46)(17,51)(18,47)(19,49)(20,27)(22,26)(23,25)(28,59)(30,35)(32,45)(33,60)(34,62)(39,55)(40,64)(42,54)(44,53)(48,50)(52,56)(57,58)(61,63);(1,5,7,26)(2,10,13,12)(3,16,19,18)(4,21,24,22)(6,25,30,58)(8,14,34,33)(9,36,38,37)(11,39,42,27)(15,46,49,47)(17,50,28,52)(20,41,55,54)(23,35,57,29)(31,43,62,60)(32,44,40,63)(45,53,64,61)(48,59,56,51)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=i,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fi,g^b=gi,e^c=ei,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307102, #72307113, #72307119, …
maximal normal subgroup of	#72307102, #72307113, #72307119, …
maximal quotient of	C₂×#493796

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400992);