#7686818

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

7375 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)(5,6,9,8)(7,11,12,10)(13,14,19,20)(15,23,24,17)(16,21,25,27)(18,22,28,26);(5,7,9,12)(6,10,8,11)(13,15,19,24)(14,17,20,23)(16,18,25,28)(21,26,27,22);(2,4)(14,20)(15,24)(16,25)(22,26);(1,2)(3,4)(13,16,19,25)(14,21,20,27)(15,18,24,28)(17,26,23,22);(5,8,9,6)(7,11,12,10)(13,17,19,23)(14,15,20,24)(16,26,25,22)(18,27,28,21);(13,18)(14,22)(15,25)(16,24)(17,21)(19,28)(20,26)(23,27)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=g,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, 7594014);