#374609

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1502 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302271, #72302279
maximal normal subgroup of	#72302271, #72302279
maximal quotient of	#72302314

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281805);