#501134

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#23915	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1157 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1157 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286439, #72286470
maximal normal subgroup of	#72286439, #72286470

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408330);