#6370993

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,27)(17,20)(18,22)(19,33)(21,30)(23,32)(25,41)(26,36)(28,44)(29,38)(31,40)(34,37)(35,43)(39,42);(6,9)(10,12)(13,15,18,17)(14,20,22,24)(16,25,29,28)(19,26,32,31)(21,35,37,39)(23,36,33,40)(27,44,38,41)(30,42,34,43);(13,16,18,29)(14,21,22,37)(15,25,17,28)(19,34,32,30)(20,35,24,39)(23,38,33,27)(26,43,31,42)(36,41,40,44);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,20,22,24)(16,28,29,25)(19,31,32,26)(21,35,37,39)(23,36,33,40)(27,44,38,41)(30,43,34,42);(15,17)(20,24)(25,28)(26,31)(35,39)(36,40)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278189);