#499115

order	512
nilpotency class	4
isoclinism	isoclinic class of #498904
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1119 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 178 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406311);