#7500840

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35527	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315405, #72315429, #72315457
maximal normal subgroup of	#72315405, #72315429, #72315457
maximal quotient of	#72315676

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408036);