#7712117

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

6906 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)(5,6,10,12)(7,15,16,11)(8,13,18,20)(9,19,17,14)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,36,35,31);(1,2)(3,4)(5,7,10,16)(6,11,12,15)(8,17,18,9)(13,19,20,14)(22,27)(23,28)(29,33)(30,35);(6,12)(7,16)(8,18)(14,19)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,36)(34,35);(5,8)(6,13)(7,17)(9,16)(10,18)(11,19)(12,20)(14,15);(5,6)(7,11)(8,13)(9,14)(10,12)(15,16)(17,19)(18,20)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,33)(34,36);(5,9,10,17)(6,14,12,19)(7,8,16,18)(11,13,15,20)(21,25,26,34)(22,29,27,33)(23,30,28,35)(24,31,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 7619313);