#7148450

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1769 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055646);