#378175

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1393 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,d^2=efhi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285371);