#6435042

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23393	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,e^c=ei,e^d=ei

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, 6342238);