#7205581

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#30549	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,33)(28,35)(29,34)(30,32)(31,37)(36,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,31)(28,38)(29,37)(30,36)(32,39)(33,34)(35,40);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(28,30)(31,39)(33,40)(34,35)(36,38);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,27)(26,33)(28,38)(29,37)(30,36)(31,34)(32,40)(35,39);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(25,28,29,30)(26,32,34,35)(27,36,37,38)(31,40,33,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fi,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=ehi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308877, #72309008, #72309019
maximal normal subgroup of	#72308877, #72309008, #72309019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7112777);