#498173

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22870
prime radical	#33538	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

723 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225388
maximal normal subgroup of	#72225388

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405369);