#353034

order	512
nilpotency class	3
isoclinism	isoclinic class of #352093
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²×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×#1721	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1135 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1135 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 256

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

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

Polycyclic presentation:

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

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, 260230);