#273024

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#37770	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180220);