#443549

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1732 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1732 conjugacy classes of subgroups — too many to draw. Only its 272 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=egi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=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, 350745);