#7698354

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

7157 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)(6,9)(10,12)(13,14)(15,20)(16,18)(17,23)(19,21)(22,27)(24,28)(25,26);(5,6)(7,10)(8,9)(11,12)(13,15,19,24)(14,20,21,28)(16,25,27,17)(18,26,22,23);(1,2,3,4)(14,21)(15,24)(16,27)(23,26);(1,2)(3,4)(13,16)(14,22)(15,25)(17,24)(18,21)(19,27)(20,23)(26,28);(5,7,8,11)(6,10,9,12)(13,17,19,25)(14,23,21,26)(15,16,24,27)(18,28,22,20);(13,18,19,22)(14,16,21,27)(15,26,24,23)(17,20,25,28)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,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, 7605550);