#378172

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 20 points:

(2,3)(5,6,9,13,10,8,15,12)(7,14,18,16,19,20,17,11);(1,2,4,3)(5,7,15,17,10,19,9,18)(6,11,12,20,8,16,13,14);(9,15)(12,13)(14,20)(17,18);(1,3,4,2)(5,8)(6,10)(7,16)(9,12)(11,19)(13,15)(14,17)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285368);