#7713102

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

6928 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)(7,11)(8,10)(9,16)(12,17)(13,14)(15,20)(18,19)(24,32)(28,33)(30,36)(34,35);(1,2,3,4)(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(6,8)(7,13)(12,16)(15,19)(22,26)(27,29)(28,33)(34,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7620298);