#235358

order	512
nilpotency class	3
isoclinism	isoclinic class of #235354
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3109	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1467 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:

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

Polycyclic presentation:

2^9|c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239668, #72249691
maximal normal subgroup of	#72239668, #72249691

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142554);