#331594

order	512
nilpotency class	3
isoclinism	isoclinic class of #331190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238790);