#108805

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

666 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=ci,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16001);