#7866501

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6475 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)(5,6,12,16)(7,13,9,11)(8,15,10,14)(17,19,18,20)(21,22,27,24)(23,31,33,28)(25,29,32,34)(26,30,36,35);(1,2)(3,4)(5,7,12,9)(6,13,16,11)(8,18,10,17)(14,19,15,20)(21,23,27,33)(22,28,24,31)(25,26,32,36)(29,35,34,30);(5,8,12,10)(6,14,16,15)(7,17,9,18)(11,19,13,20)(21,24,27,22)(23,28,33,31)(25,34,32,29)(26,35,36,30);(5,9,12,7)(6,11,16,13)(8,17,10,18)(14,20,15,19);(5,10,12,8)(6,15,16,14)(7,17,9,18)(11,19,13,20)(21,25)(22,29)(23,26)(24,34)(27,32)(28,35)(30,31)(33,36);(5,11)(6,7)(8,19)(9,16)(10,20)(12,13)(14,17)(15,18)(21,26)(22,30)(23,32)(24,35)(25,33)(27,36)(28,29)(31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7773697);