#7917249

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

6771 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)(5,6,9,10)(7,13,15,11)(8,12,17,18)(14,20,16,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,20)(17,18);(6,10)(11,13)(12,18)(19,20)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,20)(17,18)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,7,9,15)(6,11,10,13)(8,16,17,14)(12,20,18,19)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,8,9,17)(6,12,10,18)(7,14,15,16)(11,19,13,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,e^2=h,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, 7824445);