#7623194

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=g,e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224815, #72224818, #72224819, …
maximal normal subgroup of	#72224815, #72224818, #72224819, …
maximal quotient of	#72225177, #72225180, #72225185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530390);