#301904

order	512
nilpotency class	3
isoclinism	isoclinic class of #301891
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268388, #72268392, #72268402, …
maximal normal subgroup of	#72268388, #72268392, #72268402, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209100);