#142019

order	512
nilpotency class	4
isoclinism	isoclinic class of #125094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,13,15,16)(12,14)(17,18)(19,27)(20,32)(21,24)(22,36)(23,29)(25,35)(26,33)(28,40)(30,41)(31,48)(34,37)(38,46)(39,44)(42,47)(43,45);(2,5)(6,7)(9,11,12,15)(10,13)(14,16)(17,19,28,44,22,31,20,29,21,30,35,47,33,45,34,46)(18,23,37,48,26,39,25,27,24,38,32,43,36,42,40,41);(1,3,4,8)(2,6,5,7)(18,24)(23,38)(25,40)(26,36)(27,41)(32,37)(39,42)(43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=defg,c^2=ei,d^2=ghi,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, 49215);