#192933

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

967 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)(9,10,14,18,15,12,13,17)(11,19,22,16,23,24,21,20);(1,2)(3,6)(4,8)(5,7)(9,11,14,22,15,23,13,21)(10,16,18,24,12,20,17,19);(9,12)(10,15)(11,20)(13,18)(14,17)(16,23)(19,22)(21,24);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,17,12,18)(11,21,23,22)(16,19,20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342639, #72342642, #72342648
maximal normal subgroup of	#72342639, #72342642, #72342648

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100129);