#357572

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314500, #72314510, #72314536, …
maximal normal subgroup of	#72314500, #72314510, #72314536, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264768);