#482638

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#35190	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318049, #72318653, #72318754
maximal normal subgroup of	#72318049, #72318653, #72318754

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389834);