#376228

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ei,d^2=eh,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257937, #72264501, #72264535
maximal normal subgroup of	#72257937, #72264501, #72264535
maximal quotient of	#72265472, #72308389

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283424);