#413156

order	512
nilpotency class	3
isoclinism	isoclinic class of #394183
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2542 conjugacy classes of subgroups; only the 378 normal subgroups are shown.

normal subgroups only

This group has 2542 conjugacy classes of subgroups — too many to draw. Only its 378 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)(10,12)(13,14,17,20)(15,21,22,28)(16,24,25,19)(18,26,27,23);(5,6)(7,8)(13,15,16,23,17,22,25,26)(14,18,19,21,20,27,24,28);(1,2,3,4)(6,8);(9,10,11,12)(15,22)(18,27)(21,28)(23,26)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320352);