#7727968

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

6706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=hi,f^2=h,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, 7635164);