#6466209

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,19,17,20,16,15);(2,4)(6,9)(10,11)(14,19)(15,20)(16,18);(5,7,8,12)(6,10,9,11)(13,15,16,20,17,19,18,14);(13,15,16,20,17,19,18,14);(14,20)(15,19)

Polycyclic presentation:

2^9|a^2=fgi,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305396, C₂²⋊Q₈×AGL(1,ℤ/8), #72305416
maximal normal subgroup of	#72305396, C₂²⋊Q₈×AGL(1,ℤ/8), #72305416
maximal quotient of	#72305912

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373405);