#7336777

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	#30714	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 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)(3,6)(4,7)(5,8)(17,18,24,29)(19,25,30,20)(21,27,26,32)(22,28,23,31);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,29)(21,23)(22,26)(24,30)(27,31)(28,32);(9,11)(10,13)(12,16)(14,15)(17,20,22,32,24,25,23,27)(18,26,31,30,29,21,28,19);(1,4,6,5)(2,7,3,8)(10,14)(11,16)(17,21)(18,27)(19,22)(20,31)(23,30)(24,26)(25,28)(29,32);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,22,24,23)(18,28,29,31)(19,21,30,26)(20,32,25,27)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=f,e^2=fi,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, 7243973);