#476214

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

820 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383410);