#7934093

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

Polycyclic presentation:

2^9|b^2=gi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7841289);