#6374400

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,26,23,27,21,35)(19,28,31,24,29,39,36,37)(20,25,34,41,32,40,33,45)(30,42,44,38,43,48,46,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,29)(24,37)(28,39)(30,43)(31,36)(38,47)(42,48)(44,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,24)(20,30)(21,36)(22,31)(23,29)(25,38)(26,39)(27,37)(28,35)(32,43)(33,46)(34,44)(40,47)(41,48)(42,45);(2,8)(6,9)(7,16)(14,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,32)(24,38)(26,41)(27,40)(28,42)(29,43)(31,44)(35,45)(36,46)(37,47)(39,48);(20,32)(25,40)(30,43)(33,34)(38,47)(41,45)(42,48)(44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cfi,g^c=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303288, #72303297, #72303398
maximal normal subgroup of	#72303288, #72303297, #72303398

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281596);