#300472

order	512
nilpotency class	3
isoclinism	isoclinic class of #300469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62435	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207668);