#7712799

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

6974 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)(7,12)(8,13)(9,11)(10,17)(14,18)(15,20)(16,19)(25,33)(30,34)(31,36)(32,35);(1,2)(3,4)(5,7,11,8)(6,12,9,13)(10,15,18,16)(14,19,17,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,35)(32,36)(33,34);(6,9)(7,8)(14,17)(15,16)(22,27)(24,28)(30,34)(32,35);(5,8,11,7)(6,13,9,12)(10,16,18,15)(14,20,17,19)(21,23,26,29)(22,28,27,24)(25,31,33,36)(30,35,34,32);(5,9,11,6)(7,13,8,12)(10,17,18,14)(15,20,16,19)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(5,10,11,18)(6,14,9,17)(7,15,8,16)(12,19,13,20)(21,25,26,33)(22,30,27,34)(23,31,29,36)(24,32,28,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619995);