#7410033

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 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:

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

Polycyclic presentation:

2^9|a^2=fh,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316762, #72316920, #72316943, …
maximal normal subgroup of	#72316762, #72316920, #72316943, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7317229);