#227153

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	#18480	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

639 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)(3,13)(4,14)(5,9)(6,11)(7,10)(8,12)(15,16)(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);(1,3,8,15)(2,4,12,5)(6,9,7,14)(10,16,11,13)(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);(1,4,8,5)(2,3,12,15)(6,13,7,16)(9,10,14,11)(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);(1,5,8,4)(2,9,12,14)(3,10,15,11)(6,13,7,16)(17,21,23,37)(18,26,28,44)(19,31,33,46)(20,34,22,36)(24,38,40,48)(25,41,27,43)(29,42,39,47)(30,45,32,35)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^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, 134349);