#333493

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, 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

684 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 146 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)(5,6,11,15,10,7,9,13)(8,16,20,12,19,17,18,14)(23,29)(26,35)(30,36)(31,34);(5,7,10,6)(8,17,19,16)(9,13,11,15)(12,18,14,20)(21,22,24,27)(23,30,29,36)(25,33,32,28)(26,34,35,31);(2,4)(6,7)(12,14)(13,15)(16,17)(21,23,24,29)(22,26,27,35)(25,34,32,31)(28,36,33,30);(2,4)(5,8,10,19)(6,12,7,14)(9,20,11,18)(13,17,15,16)

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=i,d^2=h,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, 240689);