#472031

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285010, #72285452, #72285600
maximal normal subgroup of	#72285010, #72285452, #72285600

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379227);