#185401

order	512
nilpotency class	3
isoclinism	isoclinic class of #185255
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92597);