#301931

order	512
nilpotency class	3
isoclinism	isoclinic class of #301917
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	#62253	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1036 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)(3,9)(4,13)(5,7)(6,11)(8,10)(12,15)(14,16)(17,18,21,24)(19,22,26,30)(20,23,28,31)(25,32,27,29);(1,3,4,11,5,12,10,16)(2,6,8,9,7,14,13,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,30)(25,28)(31,32);(2,7)(6,14)(8,13)(9,15)(17,20)(18,23)(19,25,26,27)(21,28)(22,29,30,32)(24,31);(1,3,10,16,5,12,4,11)(2,6,13,15,7,14,8,9)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306407, #72306466, #72306558, …
maximal normal subgroup of	#72306407, #72306466, #72306558, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209127);