#7685644

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

6893 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)(5,6)(7,8)(9,11)(10,12)(13,14)(15,22)(16,20)(17,19)(18,21)(23,27)(24,26)(25,28);(1,2)(3,4)(13,14)(15,22)(16,20)(17,19)(18,21)(23,27)(24,26)(25,28);(5,7,10,9)(6,11,12,8)(13,15,18,17)(14,19,21,22)(16,24,25,23)(20,27,28,26);(5,8,10,11)(6,7,12,9)(13,16,18,25)(14,20,21,28)(15,23,17,24)(19,26,22,27);(5,9,10,7)(6,11,12,8)(13,14)(15,19)(16,20)(17,22)(18,21)(23,26)(24,27)(25,28);(13,17,18,15)(14,19,21,22)(16,24,25,23)(20,26,28,27)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592840);