#412818

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

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, 320014);