#7694158

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

6999 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)(5,6,9,12)(7,10,8,11)(13,14,20,19)(15,24,22,17)(16,21,27,25)(18,23,28,26);(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,26)(18,25)(19,24)(20,22)(21,28)(23,27);(2,4)(5,7,9,8)(6,10,12,11)(13,16,20,27)(14,21,19,25)(15,23,22,26)(17,18,24,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,17,20,24)(14,22,19,15)(16,18,27,28)(21,26,25,23);(5,8,9,7)(6,11,12,10)(13,18)(14,23)(15,25)(16,24)(17,27)(19,26)(20,28)(21,22);(13,19,20,14)(15,24,22,17)(16,25,27,21)(18,23,28,26)

Polycyclic presentation:

2^9|a^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, 7601354);