#7746760

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

7292 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)(15,20)(16,19)(17,21)(18,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,19,26)(14,20,16,22)(17,24,18,25)(21,27,23,28);(6,10)(8,11)(13,16,19,14)(15,20,26,22)(17,21,18,23)(24,28,25,27);(13,17,19,18)(14,21,16,23)(15,24,26,25)(20,27,22,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,22)(16,20)(17,24)(18,25)(19,26)(21,28)(23,27);(5,8,9,11)(6,7,10,12)(13,18,19,17)(14,23,16,21)(15,25,26,24)(20,28,22,27)

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 7653956);