#7940443

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

6782 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)(5,6,11,13)(7,9,8,12)(10,14,19,20)(15,17,16,18)(21,22,26,29)(23,24,31,27)(25,28,33,35)(30,32,36,34);(1,2)(3,4)(5,7,11,8)(6,12,13,9)(10,15,19,16)(14,18,20,17);(6,13)(7,8)(14,20)(15,16)(21,23,26,31)(22,27,29,24)(25,30,33,36)(28,34,35,32);(5,8,11,7)(6,9,13,12)(10,16,19,15)(14,17,20,18)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,36)(34,35);(5,9)(6,7)(8,13)(10,18)(11,12)(14,16)(15,20)(17,19)(21,24)(22,23)(25,32)(26,27)(28,30)(29,31)(33,34)(35,36);(5,10)(6,14)(7,15)(8,16)(9,17)(11,19)(12,18)(13,20)(21,25,26,33)(22,28,29,35)(23,30,31,36)(24,32,27,34)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^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, 7847639);