#233625

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1247 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)(3,12)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,45)(30,38)(32,43)(33,40)(34,42)(35,41)(37,48)(39,46)(44,47);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,39)(26,43)(27,38)(28,36)(33,48)(34,47)(35,46)(37,42)(40,45)(41,44);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,41)(27,42)(28,44)(30,48)(31,46)(32,47)(36,40)(38,39)(43,45);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,43)(25,40,42,41)(29,46,47,48)(37,45,39,44)

Polycyclic presentation:

2^9|d^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=dhi,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303170, #72303175, #72303184, …
maximal normal subgroup of	#72303170, #72303175, #72303184, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140821);