#114224

order	512
nilpotency class	3
isoclinism	isoclinic class of #114220
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,e^c=ei,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249715, #72249767
maximal normal subgroup of	#72249715, #72249767

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21420);