#392488

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1202 conjugacy classes of subgroups; only the 482 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 482 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,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, 299684);