#434410

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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	#23426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1266 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)(6,10)(11,12)(13,14,17,22,19,15,18,20)(16,21,26,28,25,23,27,24);(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,24)(17,22)(18,20)(21,27)(23,26)(25,28);(5,7,9,8)(6,11,10,12)(14,20,15,22)(16,25)(21,28,23,24)(26,27);(2,4)(5,8,9,7)(6,11,10,12)(13,16,17,26,19,25,18,27)(14,21,22,28,15,23,20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341606);