#382900

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1010 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1010 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)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,30)(29,32);(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,30)(26,31);(2,5)(6,8)(10,12)(14,15)(17,20,21,19,22,27,29,28)(18,24,26,23,25,31,32,30);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(18,25)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,g^c=gi

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, 290096);