#361164

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

1263 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)(7,11)(8,9)(10,12)(13,14)(15,26)(16,32)(17,35)(18,37)(19,24)(20,25)(21,29)(22,34)(23,36)(27,38)(28,39)(30,40)(31,42)(33,41)(43,44);(6,9)(10,11)(13,15,20,31)(14,21,25,40)(16,33,23,39)(17,28,22,41)(18,30,19,29)(24,42,37,26)(27,34,43,35)(32,38,36,44);(13,16,20,23)(14,22,25,17)(15,27,31,43)(18,35,19,34)(21,38,40,44)(24,36,37,32)(26,28,42,41)(29,33,30,39);(5,7,8,12)(6,10,9,11)(13,17,20,22)(14,23,25,16)(15,28,31,41)(18,32,19,36)(21,39,40,33)(24,34,37,35)(26,27,42,43)(29,44,30,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=ef,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,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268360);