#7712701

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,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, 7619897);