#112887

order	512
nilpotency class	3
isoclinism	isoclinic class of #112884
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

881 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257693, #72257727
maximal normal subgroup of	#72257693, #72257727
maximal quotient of	#72257797

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20083);