#417600

order	512
nilpotency class	3
isoclinism	isoclinic class of #417597
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23443	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2423 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256257, #72256285, #72262403, …
maximal normal subgroup of	#72256257, #72256285, #72262403, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324796);