#343423

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

730 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

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

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, 250619);