#7694172

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

7112 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)(6,7)(10,11)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,26)(25,27);(5,6,9,7)(8,10,12,11)(13,14,19,16)(15,22,17,20)(18,21,26,24)(23,28,25,27);(2,4)(5,7,9,6)(8,10,12,11)(13,15,19,17)(14,20,16,22)(18,23,26,25)(21,27,24,28);(1,2)(3,4)(5,8,9,12)(6,10,7,11)(13,16,19,14)(15,20,17,22)(18,24,26,21)(23,27,25,28);(13,17,19,15)(14,20,16,22)(18,23,26,25)(21,28,24,27);(13,18,19,26)(14,21,16,24)(15,23,17,25)(20,27,22,28)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=i,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, 7601368);