#271024

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
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	#36576	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1237 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:

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

Polycyclic presentation:

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

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, 178220);