#7630860

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

7826 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=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, 7538056);