#245904

order	512
nilpotency class	3
isoclinism	isoclinic class of #245903
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	#51068	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1309 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219481
maximal normal subgroup of	#72219481

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153100);