#201635

order	512
nilpotency class	3
isoclinism	isoclinic class of #201414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35144	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318638, #72318829, #72318856
maximal normal subgroup of	#72318638, #72318829, #72318856

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108831);