#435606

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319895, #72320186, #72320339
maximal normal subgroup of	#72319895, #72320186, #72320339
maximal quotient of	#72322012

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342802);