#242552

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,10,18,9,14,8,13)(7,11,15,12,17,20,16,19)(22,26)(23,30)(24,27)(29,35)(31,34)(32,33);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,15)(12,18)(13,19)(14,20)(21,22,27,35,25,29,24,26)(23,31,33,28,32,34,30,36);(2,4)(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,20)(14,19)(21,23,25,32)(22,28,29,36)(24,33,27,30)(26,34,35,31);(5,8,9,10)(6,13,14,18)(7,15,17,16)(11,12,20,19)(23,32)(28,36)(30,33)(31,34)

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dhi,d^c=dh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149748);