#7597004

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2044 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504200);