#333270

order	512
nilpotency class	3
isoclinism	isoclinic class of #333199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19698	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

635 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,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, 240466);