#319017

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

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

1094 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1094 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226213);