#395181

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1784 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255988, #72262403, #72262422, …
maximal normal subgroup of	#72255988, #72262403, #72262422, …
maximal quotient of	#72314509

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302377);