#115357

order	512
nilpotency class	3
isoclinism	isoclinic class of #115356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(5,8)(9,10)(11,21)(12,26,30,34)(13,19)(14,18)(15,20)(16,23)(17,29,35,28)(22,31,38,40)(24,32)(25,36)(27,33,39,37);(1,2)(3,5)(4,6)(7,8)(9,11,15,25)(10,16,20,32)(12,27,30,39)(13,24,14,23)(17,33,35,37)(18,36,19,21)(22,29,38,28)(26,40,34,31);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,26)(19,34)(20,35)(21,37)(23,27)(24,39)(25,38)(32,40)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22553);