#123459

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1730 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226702
maximal normal subgroup of	#72226702
maximal quotient of	#72226700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30655);