#6816984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,9,11)(7,12,8,10)(13,14)(15,20)(16,28)(17,22)(18,23)(19,24)(21,27)(25,34)(26,35)(29,38)(30,42)(31,36)(32,37)(33,39)(40,43)(41,44);(5,6)(7,10)(8,12)(9,11)(13,15,18,26)(14,20,23,35)(16,29,27,19)(17,31,32,30)(21,24,28,38)(22,36,37,42)(25,40,41,33)(34,43,44,39);(1,2,3,4)(13,16,19,15,18,27,29,26)(14,21,24,35,23,28,38,20)(17,30,33,25,32,31,40,41)(22,36,39,44,37,42,43,34);(5,7,9,8)(6,10,11,12)(17,32)(22,37)(25,41)(30,31)(33,40)(34,44)(36,42)(39,43);(5,8,9,7)(6,10,11,12)(13,17,29,40,18,32,19,33)(14,22,38,43,23,37,24,39)(15,25,16,30,26,41,27,31)(20,34,28,42,35,44,21,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724180);