#245574

order	512
nilpotency class	3
isoclinism	isoclinic class of #245567
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	C₂³×C₄	C₂⁴
prime radical	#43002	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152770);