#188801

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

845 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95997);