#272596

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

821 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,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=ghi,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, 179792);