#120195

order	512
nilpotency class	3
isoclinism	isoclinic class of #98848
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^c=eh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27391);