#6441639

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	C₂×#3055	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307437, #72307489
maximal normal subgroup of	#72307437, #72307489

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348835);