#242624

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
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

684 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149820);