#7625637

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3482	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7384 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)(7,11)(8,10)(9,12)(13,14)(15,22)(16,24)(17,19)(18,21)(20,25)(23,27)(26,28);(5,7,9,8)(6,10,12,11)(13,15,18,17)(14,19,21,22)(16,23,25,26)(20,27,24,28);(5,7,9,8)(6,11,12,10)(13,16,18,25)(14,20,21,24)(15,23,17,26)(19,27,22,28);(5,7,9,8)(6,11,12,10)(13,17,18,15)(14,19,21,22)(16,23,25,26)(20,28,24,27);(1,3,4,2)(14,21)(19,22)(20,24)(27,28);(5,8,9,7)(6,10,12,11)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532833);