#484829

order	512
nilpotency class	4
isoclinism	isoclinic class of #484583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285299, #72285719, #72285734
maximal normal subgroup of	#72285299, #72285719, #72285734

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392025);