#125550

order	512
nilpotency class	4
isoclinism	isoclinic class of #125393
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×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#95	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32)(33,34)(35,41)(36,44,45,39)(37,40)(38,42,46,48)(43,47);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35,37,43)(34,38,40,46)(36,42,45,48)(39,41,44,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,25)(22,28)(26,31)(29,32)(33,36)(34,39)(35,42)(37,45)(38,41)(40,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=cfg,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efg,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32746);