#8081035

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

7295 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)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,24,28)(16,18,22,27)(17,26,25,21);(1,2,3,4)(13,15)(14,20)(16,25)(17,22)(18,21)(19,24)(23,28)(26,27);(5,6,8,10)(7,9,11,12)(15,24)(17,25)(20,28)(21,26);(7,11)(9,12)(13,16)(14,18)(15,17)(19,22)(20,26)(21,28)(23,27)(24,25);(5,7)(6,9)(8,11)(10,12)(13,17)(14,21)(15,22)(16,24)(18,20)(19,25)(23,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,18)(14,22)(15,21)(16,23)(17,20)(19,27)(24,26)(25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=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, 7988231);