#7763287

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

7055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7670483);