#6441625

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2342 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)(15,19)(16,21)(17,22)(18,24)(20,23)(25,28)(26,27);(2,3)(5,7,8,11)(6,9,10,12)(13,15,19,14)(16,25,17,27)(18,24,20,23)(21,26,22,28);(1,2,4,3)(6,10)(9,12)(13,16,19,17)(14,21,15,22)(18,26,20,28)(23,25,24,27);(1,3,4,2)(13,17,19,16)(14,22,15,21)(18,28,20,26)(23,27,24,25);(13,18,19,20)(14,23,15,24)(16,26,17,28)(21,25,22,27)

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,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307489, #72307705, #72307760
maximal normal subgroup of	#72307489, #72307705, #72307760

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348821);