#406465

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2002	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226925, #72226928, #72226944
maximal normal subgroup of	#72226925, #72226928, #72226944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313661);