#112927

order	512
nilpotency class	3
isoclinism	isoclinic class of #112920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,d^c=dh,e^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239701, #72249987
maximal normal subgroup of	#72239701, #72249987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20123);