#245525

order	512
nilpotency class	3
isoclinism	isoclinic class of #245519
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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1352 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)(3,12)(4,7)(5,11)(6,10)(8,13)(9,16)(14,26)(15,24)(17,20)(18,22)(19,25)(21,27)(23,29)(28,32)(30,31);(2,7)(4,16)(8,12)(9,11)(10,20)(13,18)(17,29)(21,26)(22,24)(23,25)(27,30)(31,32);(4,17)(7,20)(9,23)(12,26)(13,27)(16,29)(18,30)(24,32);(1,3,5,15)(2,8,11,22)(4,18,16,13)(6,14,19,28)(7,12,9,24)(10,21,25,31)(17,30,29,27)(20,26,23,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152721);