#245108

order	512
nilpotency class	3
isoclinism	isoclinic class of #245105
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18292	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1164 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,33)(28,34)(29,35)(30,43)(31,38)(32,39)(36,41)(37,44)(40,47)(42,46)(45,48);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,22,29)(18,23,26,35)(20,27,31,41)(21,28,32,42)(24,36,38,33)(25,34,39,46)(30,40,44,48)(37,47,43,45);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,24)(19,27)(21,30)(22,31)(23,33)(25,37)(26,38)(28,40)(29,41)(32,44)(34,45)(35,36)(39,43)(42,48)(46,47);(3,10)(6,14)(11,16)(13,15)(17,21,22,32)(18,25,26,39)(19,28,29,42)(20,30,31,44)(23,34,35,46)(24,37,38,43)(27,40,41,48)(33,45,36,47)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152304);