#7631249

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

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

12870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538445);