#7901930

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

7008 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7809126);