#6343930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343584
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	C₄²	C₂⁵
prime radical	C₂×#535	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2826 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306300, #72306304, #72306309, …
maximal normal subgroup of	#72306300, #72306304, #72306309, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251126);