#113941

order	512
nilpotency class	3
isoclinism	isoclinic class of #113932
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,e^c=ei,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21137);