#6382228

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2216 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289424);