#187994

order	512
nilpotency class	3
isoclinism	isoclinic class of #187934
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	#62819	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2271 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)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(17,18,21,26,22,25,34,39)(19,27,29,40,30,23,33,36)(20,24,32,38,31,37,44,47)(28,41,43,48,42,35,45,46);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19,22,30)(18,23,25,27)(20,28,31,42)(21,33,34,29)(24,35,37,41)(26,40,39,36)(32,45,44,43)(38,48,47,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,37)(26,38)(27,41)(29,43)(30,42)(33,45)(34,44)(36,46)(39,47)(40,48);(2,8)(6,13)(7,14)(12,15)(20,31)(24,37)(28,42)(32,44)(35,41)(38,47)(43,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95190);