#7622557

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,18,14,19,28,38,15,20,29,39,30,40,50,60)(11,16,23,33,24,34,46,56,21,31,42,52,43,53,63,68)(12,17,26,36,27,37,49,59,25,35,47,57,48,58,65,70)(22,32,44,54,45,55,64,69,41,51,61,66,62,67,71,72);(2,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,32)(18,33)(19,34)(20,31)(25,41)(26,44)(27,45)(28,46)(29,42)(30,43)(35,51)(36,54)(37,55)(38,56)(39,52)(40,53)(47,61)(48,62)(49,64)(50,63)(57,66)(58,67)(59,69)(60,68)(65,71)(70,72);(1,3,4,7)(2,6,5,8)(11,21)(16,31)(22,41)(23,42)(24,43)(32,51)(33,52)(34,53)(44,61)(45,62)(46,63)(54,66)(55,67)(56,68)(64,71)(69,72);(9,12)(10,17)(11,22)(13,26)(14,27)(15,25)(16,32)(18,36)(19,37)(20,35)(21,41)(23,44)(24,45)(28,49)(29,47)(30,48)(31,51)(33,54)(34,55)(38,59)(39,57)(40,58)(42,61)(43,62)(46,64)(50,65)(52,66)(53,67)(56,69)(60,70)(63,71)(68,72);(12,25)(17,35)(22,41)(26,47)(27,48)(32,51)(36,57)(37,58)(44,61)(45,62)(49,65)(54,66)(55,67)(59,70)(64,71)(69,72)

Polycyclic presentation:

2^9|a^2=g,c^2=f,g^2=h,h^2=i,b^a=bf,c^b=ci,e^d=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529753);