#7656371

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×#3513	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6874 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)(5,6,10,14)(7,12,17,11)(8,9,13,19)(15,18,20,16)(24,32)(27,34)(30,36)(33,35);(1,2,3,4)(21,22)(23,28)(24,27)(25,29)(26,31)(30,35)(32,34)(33,36);(5,7)(6,11)(8,15)(9,16)(10,17)(12,14)(13,20)(18,19)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,8,10,13)(6,9,14,19)(7,15,17,20)(11,16,12,18)(21,24)(22,27)(23,30)(25,32)(26,33)(28,35)(29,34)(31,36);(5,7)(6,12)(8,15)(9,18)(10,17)(11,14)(13,20)(16,19)(21,23)(22,28)(24,30)(25,31)(26,29)(27,35)(32,36)(33,34);(5,9)(6,13)(7,16)(8,14)(10,19)(11,20)(12,15)(17,18)(21,22)(23,26)(24,27)(25,29)(28,31)(30,33)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563567);