#244994

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1082 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1082 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 256

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

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

Polycyclic presentation:

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

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, 152190);