#245092

order	512
nilpotency class	3
isoclinism	isoclinic class of #245084
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18284	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1409 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)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,26)(20,27)(21,28)(22,36)(23,31)(24,32)(25,33)(29,35)(30,38)(34,40)(37,39);(1,3,5,4)(2,6,8,7)(9,11,14,21)(10,15,18,28)(12,19,23,35)(13,24,25,20)(16,29,31,26)(17,32,33,27)(22,37,38,34)(30,39,36,40);(2,7)(3,4)(6,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,30)(18,31)(20,34)(21,35)(24,37)(25,38)(27,39)(28,29)(32,40)(33,36);(9,13)(10,17)(11,20)(12,22)(14,25)(15,27)(16,30)(18,33)(19,34)(21,24)(23,38)(26,39)(28,32)(29,40)(31,36)(35,37)

Polycyclic presentation:

2^9|b^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=dh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286785
maximal normal subgroup of	#72286785

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152288);