#8058697

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

7120 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,21)(16,25)(17,19)(18,22)(20,23)(24,26)(27,28);(1,2,3,4)(7,12)(10,11)(13,15,20,17)(14,21,23,19)(16,24,18,27)(22,28,25,26);(5,7)(6,10)(8,11)(9,12)(13,16,20,18)(14,22,23,25)(15,24,17,27)(19,26,21,28);(5,6,9,8)(7,11,12,10)(13,17,20,15)(14,19,23,21)(16,24,18,27)(22,28,25,26);(5,7)(6,11)(8,10)(9,12)(13,18,20,16)(14,22,23,25)(15,24,17,27)(19,28,21,26);(5,8,9,6)(7,11,12,10)(13,19)(14,15)(16,26)(17,23)(18,28)(20,21)(22,24)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965893);