#7201053

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#53014	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309019, #72309026
maximal normal subgroup of	#72309019, #72309026

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7108249);