#7672311

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

7014 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579507);