#7527837

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1968 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)(13,14,18,22)(15,19,24,34)(16,26,29,20)(17,21,32,40)(23,41,42,33)(25,35,30,38)(27,44,43,36)(28,39,31,37);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,27)(17,30)(18,24)(20,36)(21,38)(22,34)(23,28)(25,32)(26,44)(29,43)(31,42)(33,37)(35,40)(39,41);(6,10)(7,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,33)(21,35)(22,34)(26,41)(27,28)(29,42)(30,32)(31,43)(36,37)(38,40)(39,44);(6,10)(9,12)(13,16,18,29)(14,20,22,26)(15,23,24,42)(17,31,32,28)(19,33,34,41)(21,39,40,37)(25,43,30,27)(35,44,38,36);(1,3,4,2)(15,24)(19,34)(23,42)(25,30)(27,43)(33,41)(35,38)(36,44)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7435033);