#452173

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,7)(3,6,4,8)(9,10)(11,20,23,31)(12,16)(13,17,28,29)(14,18)(15,19)(21,37,24,39)(22,32)(25,33,40,38)(26,34)(27,35)(30,36);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,24)(14,29)(15,23)(16,25)(17,32)(18,31)(19,28)(20,22)(21,27)(26,38)(30,33)(34,37)(35,40)(36,39);(2,7)(6,8)(9,12,15,27)(10,16,19,35)(11,21,23,24)(13,25,28,40)(14,30,32,26)(17,33,29,38)(18,36,22,34)(20,37,31,39);(1,4,5,3)(2,8,7,6)(9,13,15,28)(10,17,19,29)(11,22,23,18)(12,25,27,40)(14,31,32,20)(16,33,35,38)(21,34,24,36)(26,37,30,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286106, #72286206, #72286224
maximal normal subgroup of	#72286106, #72286206, #72286224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359369);