#7830785

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

7100 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)(5,6,10,11)(7,12,13,8)(9,14,18,19)(15,20,17,16)(21,22,26,25)(23,27,29,31)(24,28,32,33)(30,34,35,36);(1,2)(3,4)(5,7)(6,8)(9,17)(10,13)(11,12)(14,20)(15,18)(16,19)(21,23,26,29)(22,27,25,31)(24,30,32,35)(28,34,33,36);(6,11)(7,13)(9,18)(16,20)(23,29)(27,31)(30,35)(34,36);(5,7)(6,12)(8,11)(9,15)(10,13)(14,20)(16,19)(17,18)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,8,10,12)(6,13,11,7)(9,16,18,20)(14,17,19,15)(21,25,26,22)(23,31,29,27)(24,28,32,33)(30,34,35,36);(5,9)(6,14)(7,15)(8,16)(10,18)(11,19)(12,20)(13,17)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,e^2=i,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, 7737981);