#122875

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,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, 30071);