#493535

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1786
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3127 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280034, #72280077, #72280993, …
maximal normal subgroup of	#72280034, #72280077, #72280993, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400731);