#7654588

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

6948 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)(5,6,11,16)(7,10,15,12)(8,14,9,13)(17,20,18,19)(21,22,26,24)(23,27,29,30)(25,28,33,32)(31,34,35,36);(1,2,3,4)(5,7)(6,12)(8,17)(9,18)(10,16)(11,15)(13,20)(14,19)(23,29)(27,30)(31,35)(34,36);(5,8,11,9)(6,13,16,14)(7,17,15,18)(10,19,12,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,9,11,8)(6,14,16,13)(7,18,15,17)(10,20,12,19);(5,7)(6,10)(8,17)(9,18)(11,15)(12,16)(13,19)(14,20)(21,24,26,22)(23,30,29,27)(25,28,33,32)(31,34,35,36);(5,10)(6,15)(7,16)(8,19)(9,20)(11,12)(13,18)(14,17)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,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, 7561784);