#227609

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

633 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)(3,11)(4,8)(5,12)(6,10)(7,14)(9,16)(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)(31,45,46,38,47,42,35,48);(1,3)(2,4)(5,13)(6,7)(8,16)(9,14)(10,15)(11,12)(17,19,22,32,23,33,20,30)(18,24,27,39,28,40,25,29)(21,35,36,31,37,46,34,47)(26,42,43,48,44,45,41,38);(1,4,6,15)(2,7,10,3)(5,11,16,14)(8,9,13,12)(17,20,23,22)(18,25,28,27)(19,30,33,32)(21,36,37,34)(24,29,40,39)(26,43,44,41)(31,46,47,35)(38,42,48,45);(1,4,6,15)(2,8,10,13)(3,12,7,9)(5,14,16,11)(17,21,23,37)(18,26,28,44)(19,31,33,47)(20,34,22,36)(24,38,40,48)(25,41,27,43)(29,45,39,42)(30,35,32,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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=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, 134805);