#7686137

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

7024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,e^2=i,f^2=i,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, 7593333);