#7156268

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45387	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2052 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063464);