#402096

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

901 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338539, #72338541, #72338558
maximal normal subgroup of	#72338539, #72338541, #72338558

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309292);