#298650

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1181 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205846);