#152124

order	512
nilpotency class	5
isoclinism	isoclinic class of #152123
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12697
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#127	C₂³
prime radical	#60820	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #117 ◁ #3508 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

661 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fgh,g^b=gi,e^c=ei,f^d=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59320);