#6439723

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307281, #72307447, #72307540
maximal normal subgroup of	#72307281, #72307447, #72307540

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346919);