#6459773

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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3108 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,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^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, 6366969);