#6574407

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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
prime radical	C₂×#3275	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2680 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)(11,21,22,16)(12,17,23,24)(13,20,15,18);(1,2,3,4)(5,6)(7,8)(9,11)(10,16)(12,15)(13,23)(14,22)(17,20)(18,24)(19,21);(6,8)(9,12,14,23)(10,16,19,21)(11,13,22,15)(17,20,24,18);(5,6)(7,8)(9,12)(10,17)(11,13)(14,23)(15,22)(16,18)(19,24)(20,21);(6,8)(9,13,14,15)(10,18,19,20)(11,12,22,23)(16,17,21,24)

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,e^a=ei,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

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342367, #72342368, #72342372
maximal normal subgroup of	#72342367, #72342368, #72342372

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481603);