#401628

order	512
nilpotency class	3
isoclinism	isoclinic class of #401626
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1840 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)(3,4)(5,6,10,15,11,16,9,14)(7,17,20,29,21,28,19,12)(8,22,24,13,26,31,25,32)(18,27,23,30,35,36,34,33);(5,7,9,19,11,21,10,20)(6,12,15,17,16,29,14,28)(8,23,25,18,26,34,24,35)(13,30,31,36,32,33,22,27);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,31)(15,22)(16,32)(17,33)(19,23)(20,34)(21,35)(28,30)(29,36);(2,4)(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,24,26,25)(12,28,29,17)(13,31,32,22)(18,23,35,34)(27,30,36,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238736, #72264914, #72264921, …
maximal normal subgroup of	#72238736, #72264914, #72264921, …
maximal quotient of	#72238753

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308824);