#418907

order	512
nilpotency class	3
isoclinism	isoclinic class of #417690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2263 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)(5,6,9,26,11,16,10,15)(7,17,19,12,21,14,20,29)(8,22,27,13,25,32,24,33)(18,28,23,30,35,36,34,31);(1,2,3,4)(5,7,9,19,11,21,10,20)(6,12,15,17,16,29,26,14)(8,23,24,18,25,34,27,35)(13,30,32,36,33,31,22,28);(5,8)(6,13)(7,18)(9,24)(10,27)(11,25)(12,28)(14,31)(15,32)(16,33)(17,30)(19,34)(20,23)(21,35)(22,26)(29,36);(2,4)(5,7)(6,14)(8,18)(9,20)(10,19)(11,21)(12,26)(13,31)(15,29)(16,17)(22,28)(23,24)(25,35)(27,34)(30,33)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285454, #72285534, #72285592, …
maximal normal subgroup of	#72285454, #72285534, #72285592, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326103);