#7030334

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,26)(22,25)(24,31)(27,30)(28,32);(1,2,5,6)(3,8,4,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,22)(21,23)(25,29)(26,27)(28,30)(31,32);(9,11)(10,13)(12,16)(14,15)(17,19,25,32,21,28,22,26)(18,24,29,31,23,30,20,27);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20)(18,25)(19,27)(21,29)(22,23)(24,32)(26,30)(28,31);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(18,23)(19,28)(20,29)(26,32)

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6937530);