#96264

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

710 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 210 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=g,f^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=cghi,d^b=dh

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, 3460);