#411626

order	512
nilpotency class	3
isoclinism	isoclinic class of #411625
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

1964 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251062, #72251076, #72251089
maximal normal subgroup of	#72251062, #72251076, #72251089
maximal quotient of	#72314548

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318822);