#6466145

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2202 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)(7,11)(8,9)(10,12)(13,14,19,31,18,21,17,23)(15,24,30,20,27,37,26,33)(16,22,32,42,28,35,29,36)(25,38,41,34,39,44,40,43);(2,3)(6,9)(10,11)(13,15,18,27)(14,20,21,33)(16,25,28,39)(17,30,19,26)(22,34,35,43)(23,37,31,24)(29,41,32,40)(36,44,42,38);(1,2,4,3)(5,7,8,12)(6,10,9,11)(14,21)(20,33)(22,35)(23,31)(24,37)(34,43)(36,42)(38,44);(1,3,4,2)(13,16)(14,22)(15,25)(17,29)(18,28)(19,32)(20,34)(21,35)(23,36)(24,38)(26,40)(27,39)(30,41)(31,42)(33,43)(37,44);(16,28)(22,35)(25,39)(29,32)(34,43)(36,42)(38,44)(40,41)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,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, 6373341);