#7686814

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

7312 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)(7,10)(8,9)(11,12)(13,14,19,21)(15,24,25,20)(16,17,22,27)(18,23,26,28);(1,2)(3,4)(13,15)(14,20)(16,26)(17,23)(18,22)(19,25)(21,24)(27,28);(6,9)(10,12)(14,21)(15,25)(16,22)(23,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,22)(14,17,21,27)(15,26,25,18)(20,23,24,28);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,23)(16,21)(18,20)(19,27)(24,26)(25,28);(13,18,19,26)(14,23,21,28)(15,22,25,16)(17,24,27,20)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,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, 7594010);