#418770

order	512
nilpotency class	3
isoclinism	isoclinic class of #418770
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15266
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1976 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)(3,14,18,26)(4,19,22,10)(5,11)(6,12)(7,13)(8,16,27,17)(9,21,28,23)(15,24)(20,25)(29,31)(30,32);(1,3)(2,8)(4,20)(5,17)(6,16)(7,18)(9,25)(10,29)(11,26)(12,14)(13,27)(15,21)(19,24)(22,30)(23,31)(28,32);(1,4)(2,9)(3,15)(5,21)(6,23)(7,22)(8,24)(10,12)(11,19)(13,28)(14,25)(16,20)(17,30)(18,31)(26,32)(27,29);(1,4)(2,10)(3,15)(5,23)(6,21)(7,22)(8,25)(9,12)(11,28)(13,19)(14,24)(16,30)(17,20)(18,31)(26,29)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213603, #72213695, #72213697, …
maximal normal subgroup of	#72213603, #72213695, #72213697, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325966);