#8072091

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

7247 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)(15,20)(16,21)(17,22)(18,19)(23,24)(25,27)(26,28);(1,2,3,4)(5,6,9,7)(8,10,12,11);(6,7)(8,12)(14,19)(17,21)(20,27)(24,28);(8,12)(10,11)(13,15)(14,20)(16,26)(17,24)(18,25)(19,27)(21,28)(22,23);(5,7,9,6)(8,11,12,10)(13,16,18,22)(14,21,19,17)(15,23,25,26)(20,24,27,28);(5,8,9,12)(6,10,7,11)(13,17,18,21)(14,22,19,16)(15,24,25,28)(20,23,27,26)

Polycyclic presentation:

2^9|b^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7979287);