#7732831

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

4406 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,29)(28,36)(30,32)(31,34)(33,37)(35,39)(38,40);(1,2,4,5)(3,6,7,8)(25,27)(26,31)(28,35)(29,32)(30,34)(33,39)(36,38)(37,40);(10,11)(15,17)(16,18)(22,23)(26,32)(27,34)(28,37)(38,39);(1,3,4,7)(2,6,5,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27)(26,29)(28,35)(30,34)(31,32)(33,38)(36,39)(37,40);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,27)(28,38)(30,31)(32,34)(33,35)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7640027);