#6439959

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2220 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,22)(17,23)(18,27)(19,24)(21,25)(26,28);(1,2)(3,4)(6,9)(10,12)(13,15,19,16)(14,20,24,22)(17,26,18,25)(21,23,28,27);(2,4)(13,15,19,16)(14,21,24,28)(17,25,18,26)(20,23,22,27);(2,4)(13,16,19,15)(14,22,24,20)(17,25,18,26)(21,27,28,23);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,24,27)(15,25,16,26)(20,21,22,28)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,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	#72307127, #72307249, #72307540
maximal normal subgroup of	#72307127, #72307249, #72307540

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347155);