#396358

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1433 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303554);