#242601

order	512
nilpotency class	3
isoclinism	isoclinic class of #233301
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

780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=h,d^2=h,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=dh,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, 149797);