#7625904

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×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7252 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)(5,6,9,12)(7,10,8,11)(13,14,18,17)(15,19,23,22)(16,20,26,25)(21,27,24,28);(5,6)(7,10)(8,11)(9,12);(5,7,9,8)(6,10,12,11)(14,17)(19,22)(20,25)(27,28);(13,15,18,23)(14,19,17,22)(16,24,26,21)(20,28,25,27);(13,16,18,26)(14,20,17,25)(15,21,23,24)(19,27,22,28);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,17,18,14)(15,22,23,19)(16,25,26,20)(21,28,24,27)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=h,f^2=i,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, 7533100);