#222873

order	512
nilpotency class	3
isoclinism	isoclinic class of #222720
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#121	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fhi,g^c=gh,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, 130069);