#114816

order	512
nilpotency class	3
isoclinism	isoclinic class of #114812
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	#60752	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

964 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)(4,7)(5,8)(10,14,16,13)(12,15)(17,18,22,27,23,28,21,26)(19,29,33,24,32,40,31,39)(20,25,36,43,37,44,35,42)(30,45,34,48,47,41,46,38);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,16)(13,15)(17,19)(18,24)(20,34)(21,33)(22,31)(23,32)(25,41)(26,40)(27,29)(28,39)(30,36)(35,47)(37,46)(38,42)(43,48)(44,45);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,42)(27,43)(28,44)(29,41)(31,46)(32,47)(33,34)(39,48)(40,45)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22012);