#361106

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14,20,25)(15,26,30,39)(16,31,34,41)(17,35,32,44)(18,37,19,24)(21,38,40,29)(22,28,42,33)(23,27,43,36);(5,7,9,8)(6,10,11,12)(13,15,18,29,20,30,19,38)(14,21,24,26,25,40,37,39)(16,32,33,27,34,17,28,36)(22,35,31,43,42,44,41,23);(5,8,9,7)(6,10,11,12)(13,16,18,33,20,34,19,28)(14,22,24,31,25,42,37,41)(15,27,29,17,30,36,38,32)(21,23,26,35,40,43,39,44);(6,11)(10,12)(13,17,18,36,20,32,19,27)(14,23,24,35,25,43,37,44)(15,28,29,16,30,33,38,34)(21,22,26,31,40,42,39,41)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268302);