#7625880

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂²×#553	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7776 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,f^c=fh,e^d=eh

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533076);