#7987469

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
prime radical	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7894665);