#271833

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

949 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179029);