#96304

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 860 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239280
maximal normal subgroup of	#72239280

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3500);