#390265

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

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

1127 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)(6,9)(7,11)(13,14,18,23)(15,24,28,20)(16,29,31,36)(17,19,32,22)(21,30,37,44)(25,33,27,35)(26,40,42,43)(34,39,38,41);(5,6,8,9)(7,10,11,12)(14,19)(15,25)(16,30)(17,32)(22,23)(27,28)(29,36)(31,44)(33,35)(34,40)(38,43)(39,41);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15,18,28)(14,20,23,24)(16,26,31,42)(17,27,32,25)(19,33,22,35)(21,34,37,38)(29,43,36,40)(30,41,44,39);(6,9)(10,12)(13,16,17,30,18,31,32,44)(14,21,22,36,23,37,19,29)(15,26,27,41,28,42,25,39)(20,34,35,40,24,38,33,43)

Polycyclic presentation:

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

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, 297461);