#375943

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1415 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)(8,11)(9,12)(13,15)(14,19)(16,18)(17,20);(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(1,3,4,2)(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#23996, #72245896, #72268076, …
maximal normal subgroup of	C₄×#23996, #72245896, #72268076
maximal quotient of	#72268078, #72268105, #72269836

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283139);