#7882839

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6538 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=h,f^2=i,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, 7790035);