#8070165

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

7014 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)(6,10)(7,11)(13,14,19,21)(15,20,16,18)(17,22,26,28)(23,27,24,25);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,19,16)(14,20,21,18)(17,23,26,24)(22,27,28,25);(6,10)(7,11)(14,21)(18,20)(22,28)(25,27);(7,11)(8,12)(13,16,19,15)(14,18,21,20)(17,23,26,24)(22,27,28,25);(5,7)(6,8)(9,11)(10,12)(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,27)(21,28);(5,8)(6,7)(9,12)(10,11)(13,18)(14,15)(16,21)(17,25)(19,20)(22,23)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^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, 7977361);