#186704

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#3481	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2748 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)(3,4)(5,6,9,11,8,7,12,10)(14,18)(15,16)(19,20);(2,4)(9,12)(10,11)(13,14)(15,18)(16,20)(17,19);(5,7)(6,8)(9,10)(11,12)(13,14)(15,18)(16,20)(17,19);(13,15,17,16)(14,18,19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279282, #72280121, #72309234
maximal normal subgroup of	#72279282, #72280121, #72309234
maximal quotient of	#72279611

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93900);