#494015

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#20723
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,g^b=gi,g^c=gh,g^f=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287291, #72287362, #72287450, …
maximal normal subgroup of	#72287291, #72287362, #72287450, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401211);