#373379

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

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

841 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ef,d^2=efgi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,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, 280575);