#7674169

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

10720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|c^2=hi,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316094, #72316203
maximal normal subgroup of	#72316094, #72316203

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581365);