#7673160

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

11242 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580356);