#233598

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#34643	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1232 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:

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303170, #72303184, #72303252, …
maximal normal subgroup of	#72303170, #72303184, #72303252, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140794);