#7144053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7051249);