#7696010

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

10762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=gi,e^2=i,f^2=i,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, 7603206);