#141039

order	512
nilpotency class	4
isoclinism	isoclinic class of #125094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

528 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(2,4)(5,6,7,10,8,11,12,9)(13,14)(15,23,26,38)(16,28)(17,20)(18,33)(19,25,35,29)(21,31)(22,32)(24,37)(27,42,41,39)(30,34)(36,43,44,40);(6,9)(7,12)(10,11)(13,15,24,40,32,41,30,25,17,26,31,43,18,27,16,29)(14,19,34,39,33,44,37,23,20,35,28,42,22,36,21,38);(1,2,3,4)(14,20)(19,35)(21,37)(22,33)(23,38)(28,34)(36,44)(39,42)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=ei,d^2=ghi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48235);