#230058

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=di,f^c=fh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137254);