#7148212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#45101	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7055408);