#434404

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1186 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 341600);