#375665

order	512
nilpotency class	3
isoclinism	isoclinic class of #196002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1210 conjugacy classes of subgroups — too many to draw. Only its 258 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fhi,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,h^c=hi

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, 282861);