#271025

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#36411	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1078 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=i,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, 178221);