#6438952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313045, #72313050, #72313076
maximal normal subgroup of	#72313045, #72313050, #72313076

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346148);