#7613829

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1863 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);(2,3)(6,9)(10,11)(13,14,19,23)(15,20,26,36)(16,27,29,37)(17,22,18,32)(21,31,38,28)(24,39,42,44)(25,34,30,35)(33,40,43,41);(1,2,4,3)(13,15)(14,20)(16,24)(17,30)(18,25)(19,26)(21,33)(22,35)(23,36)(27,39)(28,41)(29,42)(31,40)(32,34)(37,44)(38,43);(1,3,4,2)(13,16,17,31,19,29,18,28)(14,21,32,37,23,38,22,27)(15,24,30,40,26,42,25,41)(20,33,34,44,36,43,35,39);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,22,23,32)(15,25,26,30)(16,28,29,31)(20,34,36,35)(21,37,38,27)(24,40,42,41)(33,39,43,44)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7521025);