#234918

order	512
nilpotency class	3
isoclinism	isoclinic class of #234906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1171 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)(3,13)(4,16)(5,10)(6,11)(7,12)(8,15)(9,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,35)(28,36)(29,37)(30,39)(31,40)(32,41)(33,42)(34,43)(38,45)(44,47)(46,48);(1,3)(2,8)(4,7)(5,14)(6,15)(9,12)(10,13)(11,16)(17,19,22,29)(18,23,26,37)(20,27,31,45)(21,32,34,28)(24,38,40,35)(25,41,43,36)(30,46,33,44)(39,47,42,48);(1,4,7,3)(2,9,12,8)(5,14,6,15)(10,13,11,16)(17,20)(18,24)(19,27)(21,33)(22,31)(23,35)(25,42)(26,40)(28,46)(29,45)(30,34)(32,44)(36,48)(37,38)(39,43)(41,47);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,9,16)(17,21,22,34)(18,25,26,43)(19,28,29,32)(20,30,31,33)(23,36,37,41)(24,39,40,42)(27,44,45,46)(35,47,38,48)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=dhi,e^c=ei,f^c=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, 142114);