#6457615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2045 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)(5,6)(7,11)(8,12)(9,10)(13,14)(15,24)(16,27)(17,20)(18,22)(19,33)(21,31)(23,28)(25,39)(26,35)(29,41)(30,42)(32,38)(34,44)(36,43)(37,40);(2,3)(6,10)(11,12)(13,15,18,17)(14,20,22,24)(16,25,30,29)(19,26,28,32)(21,34,36,37)(23,35,33,38)(27,41,42,39)(31,40,43,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,28,43,18,30,19,31)(14,21,33,42,22,36,23,27)(15,25,32,44,17,29,26,40)(20,34,38,39,24,37,35,41);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,18,15)(14,20,22,24)(16,29,30,25)(19,32,28,26)(21,34,36,37)(23,35,33,38)(27,41,42,39)(31,44,43,40);(15,17)(20,24)(25,29)(26,32)(34,37)(35,38)(39,41)(40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6364811);