#285503

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 192699);