#242714

order	512
nilpotency class	3
isoclinism	isoclinic class of #242711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 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:

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

Polycyclic presentation:

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

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, 149910);