#193885

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,22,26,23,25,21,19)(20,28,30,29,31,24,32,27);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,19,21,25,23,26,22,18)(20,29,30,24,31,27,32,28);(3,9)(6,12)(10,15)(13,16)(17,20)(18,24)(19,27)(21,30)(22,32)(23,31)(25,28)(26,29);(2,7)(6,12)(8,14)(13,16)(17,21,23,22)(18,19,25,26)(20,30,31,32)(24,27,28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=ci,f^b=fh,e^c=eh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306403, #72306488, #72306492
maximal normal subgroup of	#72306403, #72306488, #72306492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101081);