#7694934

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

7123 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=h,e^2=h,f^2=h,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, 7602130);