#7920133

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

6945 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)(5,6,10,12)(7,9,14,11)(8,16,18,13)(15,17,20,19)(21,22,26,30)(23,27,33,36)(24,28,32,35)(25,29,31,34);(1,2)(3,4)(5,7,10,14)(6,11,12,9)(8,15,18,20)(13,17,16,19)(21,23)(22,27)(24,31)(25,32)(26,33)(28,34)(29,35)(30,36);(6,12)(7,14)(13,16)(15,20)(21,24)(22,28)(23,25)(26,32)(27,29)(30,35)(31,33)(34,36);(5,6)(7,11)(8,13)(9,14)(10,12)(15,17)(16,18)(19,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,34)(32,35)(33,36);(5,8)(6,13)(7,15)(9,19)(10,18)(11,17)(12,16)(14,20)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,36)(30,35);(5,9,10,11)(6,14,12,7)(8,17,18,19)(13,15,16,20)(21,25,26,31)(22,29,30,34)(23,32,33,24)(27,35,36,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,f^2=i,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, 7827329);