#102536

order	512
nilpotency class	3
isoclinism	isoclinic class of #102505
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

438 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,c^b=cf,d^b=dg,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9732);