#122874

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

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	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 257 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=h,d^2=gi,e^2=fgi,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, 30070);