#9478216

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340965
maximal subgroup of	#72340924, #72340965
maximal normal subgroup of	#72340924, #72340965

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9385412);