#7658123

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

6638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(7,13)(11,14)(15,19)(18,20)(21,22,26,27)(23,25,30,28)(24,32,34,29)(31,35,36,33);(1,2,3,4)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(22,27)(23,30)(29,32)(31,36);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,23,26,30)(22,28,27,25)(24,31,34,36)(29,35,32,33);(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,33,28,35);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,25,26,28)(22,30,27,23)(24,33,34,35)(29,36,32,31);(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=gh,c^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, 7565319);