#8075307

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

6836 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)(5,6,10,11)(7,9,13,18)(8,12,17,19)(14,16,15,20)(21,22,27,30)(23,25,28,24)(26,29,34,36)(31,33,35,32);(1,2,3,4)(21,23,27,28)(22,24,30,25)(26,31,34,35)(29,32,36,33);(6,11)(9,18)(12,19)(16,20)(21,24,27,25)(22,28,30,23)(26,33,34,32)(29,31,36,35);(5,7,10,13)(6,9,11,18)(8,15,17,14)(12,20,19,16)(21,25,27,24)(22,28,30,23)(26,32,34,33)(29,31,36,35);(5,8)(6,12)(7,14)(9,16)(10,17)(11,19)(13,15)(18,20)(21,22)(23,24)(25,28)(26,29)(27,30)(31,32)(33,35)(34,36);(5,9)(6,13)(7,11)(8,16)(10,18)(12,15)(14,19)(17,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7982503);