#244881

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1030 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152077);