#202780

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=ei,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=dh,f^c=fhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279507, #72279546, #72279570
maximal normal subgroup of	#72279507, #72279546, #72279570
maximal quotient of	#72310172, #72342560, #72342577

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109976);