#7806355

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

7112 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)(6,8)(10,12);(1,2,3,4)(6,8)(7,11)(14,19)(15,16)(17,25)(27,28);(5,6,9,8)(7,12,11,10)(13,14)(15,20)(16,21)(17,26)(18,19)(22,25)(23,28)(24,27);(5,7)(6,10)(8,12)(9,11)(13,15,18,16)(14,20,19,21)(17,23,25,24)(22,27,26,28);(5,8,9,6)(7,12,11,10)(13,16,18,15)(14,21,19,20)(17,24,25,23)(22,28,26,27);(5,7)(6,10)(8,12)(9,11)(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,27)(21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713551);