#152102

order	512
nilpotency class	5
isoclinism	isoclinic class of #152101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12696
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#60830	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3498 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 674 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)(3,14,15,29)(4,19,7,12)(5,22)(6,11)(8,13)(9,16,25,31)(10,21,27,20)(17,24,18,30)(23,28,32,26);(1,3)(2,9)(4,17)(5,23)(6,16)(7,15)(8,18)(10,14)(11,24)(12,28)(13,26)(19,25)(20,31)(21,32)(22,29)(27,30);(1,4,8,7)(2,10,13,27)(3,15,18,17)(5,21,6,20)(9,14,26,30)(11,12,22,19)(16,32,23,31)(24,28,29,25)

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59298);