#228172

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	#3480	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

641 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,e^2=h,f^2=i,g^2=h,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, 135368);