#7630840

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

8054 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)(5,6,9,12)(7,8,11,10)(13,14,20,17)(15,23,18,21)(16,26,22,19)(24,28,27,25);(5,7,9,11)(6,10,12,8)(13,15,20,18)(14,21,17,23)(16,27,22,24)(19,25,26,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16)(14,19)(15,24)(17,26)(18,27)(20,22)(21,28)(23,25);(13,17,20,14)(15,23,18,21)(16,26,22,19)(24,25,27,28);(13,18,20,15)(14,21,17,23)(16,27,22,24)(19,28,26,25);(5,8,9,10)(6,11,12,7)(13,19,20,26)(14,22,17,16)(15,25,18,28)(21,24,23,27)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=i,f^2=g,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, 7538036);