#7565985

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,3)(5,6,9,10)(7,11,8,12)(15,24)(19,25)(20,28)(26,27);(2,3)(6,10)(11,12)(13,14)(15,25)(16,23)(17,22)(18,21)(19,24)(20,26)(27,28);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,27)(17,24)(18,26)(20,23)(21,28)(22,25);(1,3,4,2)(5,7,9,8)(6,11,10,12)(13,15)(14,20)(16,26)(17,24)(18,27)(19,23)(21,25)(22,28);(5,8,9,7)(6,11,10,12)(13,16,17,18)(14,21,22,23)(15,26,24,27)(19,20,25,28)

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283563, #72283951, #72308073, …
maximal normal subgroup of	#72283563, #72283951, #72308073, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7473181);