#417800

order	512
nilpotency class	3
isoclinism	isoclinic class of #417579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2822 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283686, #72283955, #72283994, …
maximal normal subgroup of	#72283686, #72283955, #72283994, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324996);