#240225

order	512
nilpotency class	3
isoclinism	isoclinic class of #240223
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60805	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

927 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)(3,6)(4,12)(5,9)(7,11)(8,14)(10,15)(13,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,38)(30,45)(31,40)(33,44)(35,39)(36,43)(37,42)(41,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,12)(8,16)(9,15)(13,14)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,41)(26,29)(27,44)(28,40)(30,36)(34,45)(37,47)(38,46)(39,42)(43,48);(1,3)(2,7)(4,13)(5,10)(6,12)(8,15)(9,16)(11,14)(17,20,21,37,23,38,22,36)(18,25,27,42,28,34,26,43)(19,30,32,46,33,47,31,35)(24,39,29,41,40,48,44,45);(3,10)(6,15)(7,16)(11,13)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,29,40,44)(25,42,34,43)(30,46,47,35)(39,45,48,41)

Polycyclic presentation:

2^9|c^2=e,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,g^b=gh,d^c=dhi,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147421);