#192741

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1292 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)(5,6,8,11)(7,9,12,10)(14,15)(16,17)(19,20);(1,3,4,2)(6,9)(7,12)(10,11)(13,14,18,20)(15,16,19,17);(13,15)(14,16)(17,20)(18,19);(13,16,18,17)(14,19,20,15)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279172, #106×AGL(1,ℤ/8), #72279595
maximal normal subgroup of	#72279172, #106×AGL(1,ℤ/8), #72279595
maximal quotient of	#72279300, #72310098, #72310107, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99937);