#10368407

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

4472 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275603);