#357657

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂²×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3082 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213456, #72215705, #72215773, …
maximal normal subgroup of	#72213456, #72215705, #72215773, …
maximal quotient of	#72213342

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264853);