#314998

order	512
nilpotency class	3
isoclinism	isoclinic class of #252247
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	#392	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) = 64

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

(1,2,5,9)(3,6,10,14)(4,7,12,8)(11,13,16,15)(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);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(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);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(19,29)(30,43)(31,36)(45,47);(3,10)(6,14)(11,16)(13,15)(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=egi,b^2=e,d^2=e,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,d^b=di,f^b=fh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222194);