#92848

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#327
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	C₂×#336	C₂²
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂×#336 ▷ C₂³×C₄ ▷ C₂² ▷ C₁

280 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=hi,c^2=fi,d^2=g,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defi,e^b=ei,f^b=fi,d^c=dhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44);