#6439830

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2178 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347026);