#368246

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#23996, #72245877, #72245928, …
maximal normal subgroup of	C₄×#23996, #72245877, #72245928, …
maximal quotient of	#72251578, #72269846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275442);