#7625884

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×#3481	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,20)(15,18,16,19);(13,14)(15,18)(16,19)(17,20);(5,6,8,10)(7,9,11,12)(13,15,17,16)(14,18,20,19);(7,11)(9,12);(5,7)(6,9)(8,11)(10,12);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,16,17,15)(14,19,20,18)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,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.

Relationships

maximal subgroup of	#72262148, #72262558, #72262563
maximal normal subgroup of	#72262148, #72262558, #72262563

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533080);