#227151

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#3482	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

595 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,d^c=dhi,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134347);