#6434180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2128 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,20,25)(21,24,31,26)(23,30,32,29);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,20)(18,23,27,32)(21,30,31,29)(24,28,26,25);(11,15)(13,16)(18,24)(21,31)(23,28)(25,32)(26,27)(29,30);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,22,19)(18,25,27,28)(21,30,31,29)(23,24,32,26);(19,20)(23,32)(25,28)(29,30)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341376);