#7727023

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

6790 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)(8,13,15,11)(9,12,17,14)(16,19,20,18)(21,22)(23,31)(24,30)(25,28)(26,32)(27,29)(33,34)(35,36);(1,2)(3,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,23,27,25)(22,28,29,31)(24,33,32,35)(26,34,30,36);(6,7)(8,15)(12,14)(16,20)(22,29)(24,32)(28,31)(33,35);(5,7,10,6)(8,13,15,11)(9,14,17,12)(16,19,20,18)(21,24,27,32)(22,30,29,26)(23,33,25,35)(28,36,31,34);(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,25,27,23)(22,31,29,28)(24,35,32,33)(26,36,30,34);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,26,27,30)(22,32,29,24)(23,34,25,36)(28,35,31,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 7634219);