#220537

order	512
nilpotency class	3
isoclinism	isoclinic class of #220528
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3489	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127733);