#7694188

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

6700 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)(5,6,9,8)(7,11,12,10)(13,14,20,22)(15,16,17,19)(18,21,26,28)(23,24,25,27);(5,7,9,12)(6,10,8,11)(13,15,20,17)(14,19,22,16)(18,23,26,25)(21,27,28,24);(2,4)(5,6,9,8)(7,10,12,11)(13,16,20,19)(14,15,22,17)(18,24,26,27)(21,23,28,25);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,17,20,15)(14,16,22,19)(18,25,26,23)(21,24,28,27);(5,8,9,6)(7,11,12,10)(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,26)(22,28);(13,19,20,16)(14,15,22,17)(18,24,26,27)(21,25,28,23)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7601384);