#301884

order	512
nilpotency class	3
isoclinism	isoclinic class of #301861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18194	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268441, #72268442, #72268447, …
maximal normal subgroup of	#72268441, #72268442, #72268447, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209080);