#7654561

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

6885 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)(5,6,11,15)(7,13,8,12)(9,10,14,20)(16,19,18,17)(21,22,26,24)(23,27,29,30)(25,28,33,32)(31,34,35,36);(1,2)(3,4)(23,29)(27,30)(31,35)(34,36);(5,7,11,8)(6,12,15,13)(9,16,14,18)(10,17,20,19)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,8,11,7)(6,13,15,12)(9,18,14,16)(10,19,20,17);(5,9)(6,10)(7,16)(8,18)(11,14)(12,17)(13,19)(15,20)(21,24,26,22)(23,30,29,27)(25,28,33,32)(31,34,35,36);(5,10)(6,14)(7,17)(8,19)(9,15)(11,20)(12,18)(13,16)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561757);