#7897253

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

6681 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)(21,22,27,29)(23,30,24,28)(25,34,35,31)(26,32,36,33);(1,2,3,4)(21,23,27,24)(22,28,29,30)(25,32,35,33)(26,31,36,34);(6,10)(11,13)(12,17)(18,19)(22,29)(23,24)(31,34)(32,33);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24,27,23)(22,30,29,28)(25,33,35,32)(26,34,36,31);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,25,27,35)(22,31,29,34)(23,32,24,33)(26,28,36,30);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,26)(22,32)(23,31)(24,34)(25,30)(27,36)(28,35)(29,33)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=i,e^2=i,f^2=hi,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, 7804449);