#125562

order	512
nilpotency class	4
isoclinism	isoclinic class of #125508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 284 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) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=fgi,c^2=fg,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cfg,d^a=df,f^a=fi,g^a=gi,c^b=chi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32758);