#8075438

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

6547 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)(7,12,13,8)(9,14,17,19)(15,20,18,16)(21,22,27,25)(23,30,24,28)(26,29,34,33)(31,36,32,35);(1,2)(3,4)(5,6)(7,12)(8,13)(9,14)(10,11)(15,20)(16,18)(17,19);(6,11)(8,12)(9,17)(15,18)(21,23,27,24)(22,28,25,30)(26,32,34,31)(29,36,33,35);(5,7,10,13)(6,12,11,8)(9,18,17,15)(14,16,19,20)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(5,8,10,12)(6,13,11,7)(9,16,17,20)(14,18,19,15)(21,25,27,22)(23,30,24,28)(26,33,34,29)(31,36,32,35);(5,9)(6,14)(7,15)(8,16)(10,17)(11,19)(12,20)(13,18)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,e^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, 7982634);