#333528

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 142 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)(5,6,9,13,10,14,20,12)(7,15,17,8,18,19,16,11)(22,27)(28,35)(29,30)(31,36);(5,7,10,18)(6,11,14,8)(9,16,20,17)(12,15,13,19)(21,22,25,27)(23,31,33,36)(24,30,26,29)(28,34,35,32);(2,4)(5,8,10,11)(6,7,14,18)(9,19,20,15)(12,16,13,17)(21,23)(22,28)(24,32)(25,33)(26,34)(27,35)(29,31)(30,36);(2,4)(6,12)(7,16)(9,20)(13,14)(15,19)(17,18)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,31,35,36)

Polycyclic presentation:

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

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, 240724);