#390535

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

961 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=egh,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 297731);