#298656

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,10,14)(7,12,16,20)(8,17,15,11)(9,19,18,13)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,35)(34,36);(1,2)(3,4)(5,7,9,15,10,16,18,8)(6,11,13,12,14,17,19,20)(22,26)(27,34)(28,33)(29,35);(2,4)(21,23,25,31,24,30,32,36)(22,27,28,29,26,34,33,35);(5,8,18,16,10,15,9,7)(6,12,19,11,14,20,13,17)(22,26)(27,34)(28,33)(29,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=ehi,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, 205852);