#209602

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#537	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

777 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116798);