#7697605

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

8396 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)(10,11)(13,14)(15,19)(16,24)(17,21)(18,22)(20,26)(23,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,19,22,28)(16,23,26,21)(17,24,27,20);(2,4)(5,6)(7,11)(8,9)(10,12)(13,16,18,26)(14,20,22,24)(15,21,25,23)(17,19,27,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,16)(14,20)(15,23)(17,19)(18,26)(21,25)(22,24)(27,28);(13,14)(15,19)(16,20)(17,23)(18,22)(21,27)(24,26)(25,28);(13,17)(14,21)(15,24)(16,19)(18,27)(20,25)(22,23)(26,28)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,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, 7604801);