#383140

order	512
nilpotency class	3
isoclinism	isoclinic class of #213752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35088	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2150 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)(7,15,18,11)(8,12)(9,21,22,13)(10,14)(16,23,30,25)(17,28)(19,32,33,26)(20,27)(24,31)(29,34)(35,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,25)(13,28)(14,15)(19,29)(20,30)(21,24)(22,31)(23,27)(26,34)(32,35)(33,36);(2,4)(5,8,10,20)(6,12,14,27)(7,16,18,30)(9,19,22,33)(11,23,15,25)(13,26,21,32)(17,29,31,36)(24,34,28,35);(5,9)(6,13)(7,17)(8,19)(10,22)(11,24)(12,26)(14,21)(15,28)(16,29)(18,31)(20,33)(23,34)(25,35)(27,32)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251377, #72251395, #72251399, …
maximal normal subgroup of	#72251377, #72251395, #72251399, …
maximal quotient of	#72251433

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290336);