#7987103

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6283 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,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, 7894299);