#7627400

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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×#3549	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8452 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)(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,18)(17,19);(5,6,9,8)(7,11,12,10)(13,15,17,16)(14,18,19,20);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15,17,16)(14,18,19,20);(13,15,17,16)(14,18,19,20);(1,3,4,2)(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,20)(17,19);(5,8,9,6)(7,11,12,10)(13,16,17,15)(14,18,19,20)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72310326, #72310328, #72310331, …
maximal subgroup of	#72282075, #72282184, C₄×#63011, …
maximal normal subgroup of	#72282075, #72282184, C₄×#63011, …
maximal quotient of	#72282444, #72282549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534596);