#392484

order	512
nilpotency class	3
isoclinism	isoclinic class of #32929
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1266 conjugacy classes of subgroups; only the 562 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 562 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)(19,26)(20,24)(21,28)(22,25)(23,27)(29,32)(30,31);(10,13)(14,16)(17,19,23,30,22,29,21,20)(18,24,28,32,25,31,27,26);(1,2,3,4)(18,25)(24,31)(26,32)(27,28);(9,11,12,15)(10,14,13,16)(17,20,21,29,22,30,23,19)(18,26,27,31,25,32,28,24)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fi,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299680);