#7324976

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324436
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	#44884	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7232172);