#182808

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=i,e^2=h,g^2=hi,b^a=be,c^a=ch,e^a=eh,g^a=ghi,d^b=dh,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279203, #72280092, #72280136
maximal normal subgroup of	#72279203, #72280092, #72280136
maximal quotient of	#72279558

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90004);