#6914600

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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)(3,6)(4,8)(5,7)(17,18,23,20)(19,26,30,32)(21,25,22,31)(24,28,27,29);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,20)(19,27,30,24)(21,31,22,25)(26,29,32,28);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,28,23,30,22,29)(18,24,31,32,20,27,25,26);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,20,23,18)(19,24,30,27)(21,25,22,31)(26,28,32,29);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,23,22)(18,25,20,31)(19,28,30,29)(24,26,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6821796);