#228269

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	C₂×#41	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

617 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^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=dehi,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, 135465);