#6440823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2744 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348019);