#242364

order	512
nilpotency class	3
isoclinism	isoclinic class of #10611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#396
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

596 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149560);