#212840

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 258 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120036);