#123019

order	512
nilpotency class	4
isoclinism	isoclinic class of #123011
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#12102	C₂
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

351 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 64

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

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

Polycyclic presentation:

2^9|c^2=hi,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=eh,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30215);