#6484957

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,24,28)(19,22,27,23)(20,25,21,26)(29,32,30,31);(2,5)(6,7)(10,14)(11,12)(17,19,23,18,24,27,22,28)(20,30,31,26,21,29,32,25);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,24)(22,32)(23,31)(26,28)(27,30);(9,12,13,11)(10,15,14,16)(17,21,24,20)(18,26,28,25)(19,30,27,29)(22,31,23,32);(10,14)(15,16)(17,22,24,23)(18,19,28,27)(20,31,21,32)(25,30,26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6392153);