#247031

order	512
nilpotency class	3
isoclinism	isoclinic class of #247017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1231 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1231 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154227);