#188011

order	512
nilpotency class	3
isoclinism	isoclinic class of #187928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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	#62781	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2218 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,21,26,22,25,34,39)(19,27,30,40,29,23,33,36)(20,24,32,38,31,37,44,47)(28,35,43,46,42,41,45,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,23)(20,28)(21,33)(22,29)(24,35)(25,27)(26,40)(30,34)(31,42)(32,45)(36,39)(37,41)(38,48)(43,44)(46,47);(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,37)(26,38)(27,41)(29,42)(30,43)(33,45)(34,44)(36,46)(39,47)(40,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,25)(23,27)(24,37)(26,39)(35,41)(36,40)(38,47)(46,48)

Polycyclic presentation:

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

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, 95207);