#346369

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3497	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3334 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 3334 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei

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, 253565);