#6637135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636908
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,31,24)(20,21,25,23)(28,29,32,30);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,24,26,27)(20,32,25,28)(21,29,23,30);(11,15)(13,16)(17,20,23,18,22,25,21,26)(19,28,30,27,31,32,29,24);(2,6)(3,8)(11,15)(13,16)(17,21,22,23)(18,20,26,25)(19,29,31,30)(24,32,27,28);(2,6)(5,7)(10,14)(13,16)(17,21,22,23)(18,25,26,20)(19,30,31,29)(24,32,27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=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, 6544331);