#6343618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343406
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=hi,f^2=i,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318393, #72318842, #72318860
maximal normal subgroup of	#72318393, #72318842, #72318860

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250814);