#123503

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1802 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303572
maximal normal subgroup of	#72303572
maximal quotient of	#72303629

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30699);