#385598

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

919 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292794);