#7697516

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

7463 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,25)(17,21)(18,22)(20,26)(23,27)(24,28);(1,2)(3,4)(13,15,18,24)(14,19,22,28)(16,23,26,17)(20,21,25,27);(6,9)(10,12)(13,16,18,26)(14,20,22,25)(15,17,24,23)(19,27,28,21);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,23)(17,24)(18,26)(19,21)(22,25)(27,28);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,20)(17,27)(18,22)(21,23)(24,28)(25,26);(13,17,18,23)(14,21,22,27)(15,16,24,26)(19,25,28,20)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=hi,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, 7604712);