#7330704

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#30846	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,d^2=f,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^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7237900);