#7633226

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

8210 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)(13,14,20,18)(15,21,24,19)(16,25,17,22)(23,27,26,28);(1,2)(3,4)(13,15)(14,21)(16,26)(17,23)(18,19)(20,24)(22,27)(25,28);(5,6)(7,11)(8,12)(9,10)(13,16,20,17)(14,22,18,25)(15,23,24,26)(19,27,21,28);(5,7,9,8)(6,11,10,12)(13,17,20,16)(14,22,18,25)(15,23,24,26)(19,28,21,27);(5,7,9,8)(6,11,10,12)(13,18,20,14)(15,19,24,21)(16,25,17,22)(23,27,26,28);(5,8,9,7)(6,12,10,11)(13,19)(14,15)(16,27)(17,28)(18,24)(20,21)(22,23)(25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540422);