#6574375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2733 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342367, #72342369, #72342373
maximal normal subgroup of	#72342367, #72342369, #72342373

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481571);