#185210

order	512
nilpotency class	3
isoclinism	isoclinic class of #161128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2466 conjugacy classes of subgroups; only the 354 normal subgroups are shown.

normal subgroups only

This group has 2466 conjugacy classes of subgroups — too many to draw. Only its 354 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 — 4 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)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,26,32)(22,27,29,36)(24,31,25,34)(28,35,33,30);(2,4)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30);(22,29)(27,36)(28,33)(30,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92406);