#227607

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
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	#18494	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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=de,e^b=eh,d^c=dhi,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134803);