#246774

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46133	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1127 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153970);