#7671289

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

7351 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)(6,10)(11,12)(13,14)(15,19)(16,21)(17,25)(18,20)(22,26)(23,28)(24,27);(5,6)(7,11)(8,12)(9,10)(13,15,18,16)(14,19,20,21)(17,23,26,24)(22,27,25,28);(2,4)(14,20)(15,16)(22,25)(23,24);(1,2)(3,4)(13,16,18,15)(14,21,20,19)(17,24,26,23)(22,28,25,27);(5,7,9,8)(6,11,10,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,28)(25,26);(5,8,9,7)(6,12,10,11)(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,25)(21,28)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=g,f^2=g,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, 7578485);