#6470095

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#546	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310179
maximal normal subgroup of	#72310179
maximal quotient of	#72310209

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377291);