#469684

order	512
nilpotency class	3
isoclinism	isoclinic class of #460131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1273 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283557, #72283657, #72283918, …
maximal normal subgroup of	#72283557, #72283657, #72283918, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376880);