#234284

order	512
nilpotency class	3
isoclinism	isoclinic class of #10320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	#60734	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1670 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310568, #72310569, #72310573, …
maximal normal subgroup of	#72310568, #72310569, #72310573, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141480);