#7064921

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1789 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,22,31,17)(15,24,26,34,27,37,32,20)(16,21,30,35,29,36,41,28)(25,38,40,43,39,44,42,33);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,22,34)(16,25,29,39)(17,24,23,37)(18,32,31,26)(21,33,36,43)(28,38,35,44)(30,42,41,40);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,28)(18,30)(19,29)(20,33)(22,36)(23,35)(24,38)(26,40)(27,39)(31,41)(32,42)(34,43)(37,44);(7,12)(9,11)(13,17,31,22,19,23,18,14)(15,20,32,37,27,34,26,24)(16,28,41,36,29,35,30,21)(25,33,42,44,39,43,40,38);(6,10)(9,11)(14,22)(16,29)(17,23)(20,34)(24,37)(25,39)(30,41)(40,42)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 6972117);