#7614174

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1727 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521370);