#7638594

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545790);