#7876829

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

7638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=hi,e^2=h,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, 7784025);