#412223

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 319419);