#7053974

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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	#58999	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961170);