#348163

order	512
nilpotency class	3
isoclinism	isoclinic class of #348158
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3460	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1045 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1045 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,c^b=ch,f^c=fh,g^c=gi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255359);