#7658035

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

6740 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)(7,13)(11,14)(15,19)(18,20)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33);(1,2)(3,4)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=i,e^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, 7565231);