#7932386

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

7035 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7839582);