#7883019

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

7347 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)(5,6)(7,8)(9,10)(11,12)(13,14,18,17)(15,19,21,23)(16,20,24,25)(22,26,27,28);(1,2)(3,4)(5,7,11,9)(6,8,12,10)(15,21)(19,23)(22,27)(26,28);(5,8,11,10)(6,9,12,7)(13,15,18,21)(14,19,17,23)(16,22,24,27)(20,26,25,28);(5,7,11,9)(6,8,12,10);(5,9,11,7)(6,8,12,10)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(5,10,11,8)(6,9,12,7)(13,17,18,14)(15,23,21,19)(16,20,24,25)(22,26,27,28)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790215);