#381174

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 250 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:

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

Polycyclic presentation:

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

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, 288370);