#6465593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72322498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372789);