#6446631

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4098 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)(7,11)(8,9)(10,12)(13,14,18,22)(15,23,26,37)(16,28,29,20)(17,31,24,21)(19,32,34,27)(25,39,40,36)(30,35,38,43)(33,41,42,44);(1,2)(3,4)(6,9)(10,11)(13,15,24,32,18,26,17,27)(14,19,31,37,22,34,21,23)(16,25,38,44,29,40,30,41)(20,33,43,39,28,42,35,36);(2,4)(5,7,8,12)(6,10,9,11)(13,16,18,29)(14,20,22,28)(15,25,26,40)(17,30,24,38)(19,33,34,42)(21,35,31,43)(23,36,37,39)(27,41,32,44);(2,4)(16,29)(20,28)(25,40)(30,38)(33,42)(35,43)(36,39)(41,44);(15,26)(19,34)(23,37)(25,40)(27,32)(33,42)(36,39)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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, 6353827);