#400719

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
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	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1453 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)(8,18)(9,20)(12,22)(13,23)(14,24)(16,31)(25,35)(26,34)(27,28)(29,32)(33,36);(1,2)(3,4)(5,6,9,11,10,14,20,24)(7,12,16,22,17,26,31,34)(8,13,21,23,18,28,19,27)(15,25,32,33,30,36,29,35);(2,4)(5,7)(6,12)(8,15)(9,16)(10,17)(11,22)(13,25)(14,26)(18,30)(19,29)(20,31)(21,32)(23,33)(24,34)(27,35)(28,36);(5,8,9,21,10,18,20,19)(6,13,11,23,14,28,24,27)(7,15,16,32,17,30,31,29)(12,25,22,33,26,36,34,35)

Polycyclic presentation:

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

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, 307915);