#6911150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#24425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1634 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6818346);