#417132

order	512
nilpotency class	3
isoclinism	isoclinic class of #394329
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1480 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1480 conjugacy classes of subgroups — too many to draw. Only its 266 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fg,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324328);