#152762

order	512
nilpotency class	5
isoclinism	isoclinic class of #149927
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#535 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,9,13,11,15,8,23)(7,17,21,36,20,12,28,31)(10,14,24,29,27,16,26,33)(18,35,19,32,25,30,22,34);(5,7,9,25)(6,12,29,36)(8,18,11,20)(10,22,24,21)(13,30,16,34)(14,32,23,35)(15,17,33,31)(19,26,28,27);(2,4)(5,8)(6,13)(7,18)(9,11)(10,24)(12,30)(14,33)(15,23)(16,29)(17,35)(19,28)(20,25)(21,22)(26,27)(31,32)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284670, #72284690, #72286041
maximal normal subgroup of	#72284670, #72284690, #72286041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59958);