#7593006

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

1744 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)(13,14,19,24)(15,22,26,40)(16,21,28,36)(17,31,27,20)(18,34,33,23)(25,39,41,37)(29,35,32,44)(30,43,42,38);(2,3)(5,7)(6,9)(8,11)(10,12)(13,15,18,27,19,26,33,17)(14,20,23,22,24,31,34,40)(16,25,30,29,28,41,42,32)(21,35,38,39,36,44,43,37);(1,2,4,3)(13,16,19,28)(14,21,24,36)(15,25,26,41)(17,32,27,29)(18,30,33,42)(20,35,31,44)(22,39,40,37)(23,38,34,43);(1,3,4,2)(7,11)(9,12)(15,26)(16,28)(17,27)(20,31)(21,36)(22,40)(30,42)(38,43);(13,17,33,26,19,27,18,15)(14,22,34,20,24,40,23,31)(16,29,42,25,28,32,30,41)(21,37,43,44,36,39,38,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,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,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, 7500202);