#6465589

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(5,6,9,14,11,15,10,25)(7,16,19,28,20,27,18,12)(8,21,22,29,24,35,23,13)(17,26,32,30,33,36,31,34);(2,3)(5,7)(6,12)(8,17)(9,18)(10,19)(11,20)(13,26)(14,27)(15,28)(16,25)(21,34)(22,31)(23,32)(24,33)(29,36)(30,35);(1,2,4,3)(5,8,10,23,11,24,9,22)(6,13,14,21,15,29,25,35)(7,17,19,32,20,33,18,31)(12,26,27,34,28,36,16,30);(1,3,4,2);(5,9,11,10)(6,14,15,25)(7,18,20,19)(8,22,24,23)(12,27,28,16)(13,21,29,35)(17,31,33,32)(26,34,36,30)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372785);