#96261

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

1098 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 218 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,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, 3457);