#7727019

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

6950 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)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(21,22)(23,26)(24,28)(25,33)(27,29)(30,34)(31,36)(32,35);(1,2,3,4)(21,23,27,24)(22,28,29,26)(25,31,34,32)(30,36,33,35);(6,7)(11,13)(12,14)(18,19)(22,29)(23,24)(25,34)(35,36);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,24,27,23)(22,28,29,26)(25,32,34,31)(30,36,33,35);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,25)(22,30)(23,31)(24,32)(26,35)(27,34)(28,36)(29,33);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,26,27,28)(22,23,29,24)(25,35,34,36)(30,31,33,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

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, 7634215);