#188760

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

912 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95956);