#271457

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
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	C₂×C₄²	C₂⁴
prime radical	#36160	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1168 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)(7,11)(8,10)(9,13)(12,16)(14,17)(15,19)(18,20)(21,22,26,28,25,30,24,29)(23,31,32,27,34,36,33,35);(1,2)(3,4)(6,10)(8,16)(11,17)(12,13)(14,20)(18,19);(2,4)(5,7,9,15)(6,11,13,19)(8,14,16,20)(10,17,12,18)(21,23)(22,27)(24,32)(25,34)(26,33)(28,31)(29,36)(30,35);(6,10)(8,16)(11,17)(12,13)(14,20)(18,19)(21,24,25,26)(22,28,30,29)(23,32,34,33)(27,31,35,36)

Polycyclic presentation:

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

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, 178653);