#7689868

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

7379 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)(6,10)(11,12)(13,14,19,21)(15,18,24,20)(16,22,25,27)(17,23,26,28);(5,6,9,10)(7,11,8,12)(13,15,19,24)(14,20,21,18)(16,17,25,26)(22,28,27,23);(2,4)(5,7,9,8)(6,11,10,12)(14,21)(15,24)(16,25)(23,28);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16,19,25)(14,22,21,27)(15,17,24,26)(18,23,20,28);(5,8,9,7)(6,12,10,11)(13,17)(14,23)(15,25)(16,24)(18,27)(19,26)(20,22)(21,28);(13,18,19,20)(14,24,21,15)(16,23,25,28)(17,22,26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=g,e^2=gi,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, 7597064);