#6345413

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60536	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ehi,f^a=fh,f^b=fh,d^c=dhi,e^c=eh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306799, #72306808, #72306814, …
maximal normal subgroup of	#72306799, #72306808, #72306814, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252609);