#7696631

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

7236 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)(15,21,16,17)(18,22,27,28)(23,26,24,25);(5,7,9,12)(6,10,8,11)(13,15,19,16)(14,17,20,21)(18,23,27,24)(22,25,28,26);(2,4)(5,6,9,8)(7,10,12,11)(14,20)(15,16)(22,28)(23,24);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,16,19,15)(14,21,20,17)(18,24,27,23)(22,26,28,25);(5,8,9,6)(7,11,12,10)(13,17,19,21)(14,15,20,16)(18,26,27,25)(22,24,28,23);(13,18)(14,22)(15,23)(16,24)(17,25)(19,27)(20,28)(21,26)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603827);