#245307

order	512
nilpotency class	3
isoclinism	isoclinic class of #245303
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61679	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1095 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152503);