#184018

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=chi,d^a=dh,e^a=eh,c^b=ci,e^b=eh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287402, #72287437, #72287440, …
maximal normal subgroup of	#72287402, #72287437, #72287440, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91214);