#6464633

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ 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) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371829);