#7647025

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

6664 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)(7,13,9,11)(8,12,18,16)(15,20,17,19)(21,22,27,28)(23,26,30,25)(24,32,35,29)(31,33,36,34);(1,2,3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,19)(16,18)(17,20)(22,28)(23,30)(29,32)(31,36);(5,7,10,9)(6,11,14,13)(8,15,18,17)(12,19,16,20)(21,23,27,30)(22,25,28,26)(24,31,35,36)(29,33,32,34);(5,6)(7,11)(8,16)(9,13)(10,14)(12,18)(15,20)(17,19)(21,24,27,35)(22,29,28,32)(23,31,30,36)(25,33,26,34);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16)(21,25,27,26)(22,23,28,30)(24,33,35,34)(29,31,32,36);(5,9,10,7)(6,13,14,11)(8,17,18,15)(12,20,16,19)(21,26,27,25)(22,30,28,23)(24,34,35,33)(29,36,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7554221);