#115145

order	512
nilpotency class	3
isoclinism	isoclinic class of #100203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#120	C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1618 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1618 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,g^b=gh,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22341);