#148290

order	512
nilpotency class	4
isoclinism	isoclinic class of #135848
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249918, #72249926, #72249962, …
maximal normal subgroup of	#72249918, #72249926, #72249962, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55486);