#7625914

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	C₂×#3480	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=h,e^2=h,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, 7533110);