#135973

order	512
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

617 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43169);