#403954

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#3490	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261090, #72261540, #72284362, …
maximal normal subgroup of	#72261090, #72261540, #72284362, …
maximal quotient of	#72261169, #72261645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311150);