#395182

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1856 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)(8,20)(9,12)(13,22)(14,29)(15,24)(16,30)(17,18)(21,25)(27,28)(31,32)(33,36);(1,2)(3,4)(5,6,12,24,10,15,9,11)(7,13,18,27,19,28,17,22)(8,14,25,34,21,29,20,23)(16,26,30,36,32,35,31,33);(2,4)(5,7)(6,13)(8,16)(9,17)(10,19)(11,22)(12,18)(14,26)(15,28)(20,31)(21,32)(23,33)(24,27)(25,30)(29,35)(34,36);(5,8,9,20,10,21,12,25)(6,14,11,23,15,29,24,34)(7,16,17,31,19,32,18,30)(13,26,22,33,28,35,27,36)

Polycyclic presentation:

2^9|b^2=ei,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, #72262411, #72262421, …
maximal normal subgroup of	#72255988, #72262411, #72262421, …
maximal quotient of	#72314548

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302378);