#115146

order	512
nilpotency class	3
isoclinism	isoclinic class of #100203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1198 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1198 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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,g^b=gh,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22342);