#7736314

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
prime radical	C₂×#3514	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6921 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)(5,6,10,13)(7,11,14,8)(9,12,18,20)(15,19,17,16)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34);(1,2)(3,4)(5,7)(6,11)(8,13)(9,17)(10,14)(12,16)(15,18)(19,20)(22,26)(24,32)(27,29)(30,36);(7,14)(8,11)(15,17)(16,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8,10,11)(6,7,13,14)(9,16,18,19)(12,15,20,17);(5,7)(6,11)(8,13)(9,15)(10,14)(12,19)(16,20)(17,18)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,20)(14,17)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7643510);