#314966

order	512
nilpotency class	3
isoclinism	isoclinic class of #251791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

463 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=i,d^2=ei,e^2=h,f^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,f^c=fi

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, 222162);