#499108

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	#37187	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1175 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Relationships

maximal subgroup of	#72286439, #72286626, #72286818
maximal normal subgroup of	#72286439, #72286626, #72286818

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406304);