#7737809

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

7350 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7645005);