#122857

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×#331
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

305 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=gi,e^2=fg,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30053);