#315012

order	512
nilpotency class	3
isoclinism	isoclinic class of #251818
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

416 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)(3,7)(4,12)(5,14)(6,10)(8,13)(9,15)(11,16)(17,18,23,26,22,27,21,35)(19,28,36,39,29,42,31,24)(20,32,37,40,33,46,34,25)(30,44,47,48,43,38,45,41);(1,3)(2,7)(4,6)(5,11)(8,9)(10,12)(13,15)(14,16)(17,19,21,31,22,29,23,36)(18,24,26,28,27,39,35,42)(20,30,34,45,33,43,37,47)(25,38,32,41,40,44,46,48);(1,4,6,3)(2,8,10,16)(5,11,15,13)(7,14,12,9)(19,29)(20,33)(25,40)(31,36)(32,46)(34,37)(38,44)(41,48);(3,4)(7,12)(8,16)(11,13)(17,20,21,34,22,33,23,37)(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,c^2=i,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,d^c=dhi,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222208);