#7813149

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

4546 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7720345);