#392536

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
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×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

870 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 330 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) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ef,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,c^b=ci,d^c=di

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, 299732);