#7689163

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

7448 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)(6,9)(10,12)(13,14,18,20)(15,19,22,16)(17,21,26,27)(23,25,28,24);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,20)(17,23)(18,22)(21,25)(24,27)(26,28);(2,4)(14,20)(15,22)(21,27)(23,28);(1,2)(3,4)(13,15)(14,19)(16,20)(17,23)(18,22)(21,25)(24,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,15,20,22)(17,25,26,24)(21,28,27,23);(13,17,18,26)(14,21,20,27)(15,23,22,28)(16,24,19,25)

Polycyclic presentation:

2^9|a^2=i,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, 7596359);