#352197

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

816 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ch,d^b=di,d^c=dh

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, 259393);