#142458

order	512
nilpotency class	4
isoclinism	isoclinic class of #142216
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

721 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,10,13)(7,14,16,19)(8,12,9,17)(11,18,20,15)(21,22,25,28)(23,29,31,36)(24,27,32,35)(26,33,34,30);(5,7,10,16)(6,11,13,20)(8,15,9,18)(12,19,17,14)(21,23)(22,26)(24,30)(25,31)(27,29)(28,34)(32,33)(35,36);(2,4)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(24,32)(27,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,f^d=fi

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, 49654);