#7592977

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 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)(5,6,8,10)(7,9,11,12)(13,14)(15,22)(16,21)(17,32)(18,33)(19,24)(20,28)(23,25)(26,36)(27,38)(29,35)(30,42)(31,43)(34,40)(37,39)(41,44);(2,3)(5,7)(6,9)(8,11)(10,12)(13,15,25,17,19,27,18,28)(14,20,33,38,24,32,23,22)(16,26,39,30,29,41,31,40)(21,34,43,44,35,42,37,36);(1,2,4,3)(13,16,19,29)(14,21,24,35)(15,26,27,41)(17,30,28,40)(18,31,25,39)(20,34,32,42)(22,36,38,44)(23,37,33,43);(1,3,4,2)(7,11)(9,12)(15,27)(16,29)(17,28)(20,32)(21,35)(22,38)(31,39)(37,43);(13,17,25,27,19,28,18,15)(14,22,33,20,24,38,23,32)(16,30,39,41,29,40,31,26)(21,36,43,34,35,44,37,42)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=gi,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,e^b=ei,d^c=di,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, 7500173);