#8091789

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

7097 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)(7,10)(8,9)(11,12)(13,14,18,21)(15,16,23,19)(17,20,27,28)(22,24,25,26);(1,2,3,4);(7,11)(10,12)(13,15,18,23)(14,19,21,16)(17,25,27,22)(20,24,28,26);(6,9)(7,11)(13,16)(14,15)(17,26)(18,19)(20,25)(21,23)(22,28)(24,27);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,22)(18,23)(19,21)(20,24)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,17)(14,20)(15,22)(16,24)(18,27)(19,26)(21,28)(23,25)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7998985);