#7746749

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

7126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7653945);