#400625

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(2,3)(5,6,9,12)(7,11,8,10)(13,14,17,21)(15,24,16,19)(18,22,28,23)(20,27,25,26);(1,2,4,3)(5,7,9,8)(6,10,12,11)(13,14,17,21)(15,25,16,20)(18,23,28,22)(19,27,24,26);(1,3,4,2)(5,7,9,8)(6,11,12,10)(13,15,17,16)(14,19,21,24)(18,27,28,26)(20,23,25,22);(5,8,9,7)(6,10,12,11)(13,16,17,15)(14,20,21,25)(18,27,28,26)(19,23,24,22)

Polycyclic presentation:

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

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, 307821);