#298674

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
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×#3169	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1208 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205870);