#494018

order	512
nilpotency class	4
isoclinism	isoclinic class of #480315
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#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 ◁ C₄ ◁ #97 ◁ #2875 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

750 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)(7,14,21,29)(13,24,16,22)(17,26)(18,25)(19,27)(20,28,23,30)(31,32);(1,3)(2,7,11,21)(4,16)(5,14)(6,15)(8,10,12,27)(9,23,26,20)(13,17)(18,30)(19,29)(22,32,24,25)(28,31);(2,8)(4,17)(7,10)(9,24)(11,12)(13,16)(18,31)(20,32)(21,27)(22,26)(23,25)(28,30);(1,4,5,18,6,17,19,31)(2,9,10,25,11,26,27,32)(3,13,14,28,15,16,29,30)(7,20,12,24,21,23,8,22)

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,d^b=di,g^b=gi,d^c=di,g^c=gh,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231624, #72231703, #72231719, …
maximal normal subgroup of	#72231624, #72231703, #72231719, …
maximal quotient of	#72240802, C₂×#494018, #72313669

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401214);