#8063887

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

7773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 7971083);