#233571

order	512
nilpotency class	3
isoclinism	isoclinic class of #232760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#34963	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238993, #72249767, #72279567, …
maximal normal subgroup of	#72238993, #72249767, #72279567, …
maximal quotient of	C₂×#233571

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140767);