#7686449

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
prime radical	#34681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7057 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7593645);