#432899

order	512
nilpotency class	3
isoclinism	isoclinic class of #427387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1140 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283616, #72283657, #72283895, …
maximal normal subgroup of	#72283616, #72283657, #72283895, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340095);