#195037

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#2433	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1327 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102233);