#148299

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

658 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)(5,6,11,17,10,15,12,16)(7,18,21,29,20,30,14,9)(8,22,27,33,25,13,26,31)(19,35,23,28,36,34,32,24);(5,7,9,15,12,21,30,16,10,20,29,6,11,14,18,17)(8,23,24,31,26,36,34,22,25,32,28,33,27,19,35,13);(2,4)(5,8)(6,13)(7,19)(9,28)(10,25)(11,27)(12,26)(14,32)(15,22)(16,33)(17,31)(18,34)(20,36)(21,23)(24,29)(30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250815, #72250823, #72250826, …
maximal normal subgroup of	#72250815, #72250823, #72250826, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55495);