#383863

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1545 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)(7,9)(8,10)(11,12)(13,14,18,21,19,15,22,24)(16,17,26,25,20,23,27,28);(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,25)(17,26)(18,21)(20,28)(22,24)(23,27);(6,10)(9,12)(13,16,19,20)(14,17,15,23)(18,26,22,27)(21,25,24,28);(6,10)(9,12)(13,17)(14,20)(15,16)(18,28)(19,23)(21,26)(22,25)(24,27)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291059);