#188023

order	512
nilpotency class	3
isoclinism	isoclinic class of #187949
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62820	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2282 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,22,25,32,38)(19,23,29,35,28,34,42,46)(20,24,31,37,30,36,43,47)(27,39,41,48,40,33,44,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,34)(26,35)(30,40)(31,41)(32,42)(36,39)(37,45)(38,46)(43,44)(47,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,36)(26,37)(28,40)(29,41)(32,43)(34,39)(35,45)(38,47)(42,44)(46,48);(2,8)(4,12)(6,9)(10,16)(18,25)(23,34)(24,36)(26,38)(33,39)(35,46)(37,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234502, #72234937, #72234938, …
maximal normal subgroup of	#72234502, #72234937, #72234938, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95219);