#147572

order	512
nilpotency class	4
isoclinism	isoclinic class of #129424
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#35149	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1755 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1755 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) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281001
maximal normal subgroup of	#72281001
maximal quotient of	#72281757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54768);