#6451988

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2705 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)(7,12)(8,10)(9,11)(13,14,17,22,18,21,30,35)(15,23,25,32,26,36,29,19)(16,20,28,34,27,33,40,43)(24,37,39,42,38,44,41,31);(1,2)(3,4)(13,15,18,26)(14,19,21,32)(16,24,27,38)(17,29,30,25)(20,31,33,42)(22,36,35,23)(28,41,40,39)(34,44,43,37);(2,4)(5,7,9,8)(6,10,11,12)(13,16,18,27)(14,20,21,33)(15,24,26,38)(17,28,30,40)(19,31,32,42)(22,34,35,43)(23,37,36,44)(25,39,29,41);(2,4)(5,8,9,7)(6,10,11,12)(16,27)(20,33)(24,38)(28,40)(31,42)(34,43)(37,44)(39,41);(14,21)(19,32)(20,33)(22,35)(23,36)(31,42)(34,43)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359184);